Coding and modulation apparatus using non-uniform constellation

ABSTRACT

A coding and modulation apparatus and method are presented. The apparatus comprises an encoder that encodes input data into cell words, and a modulator that modulates said cell words into constellation values of a non-uniform constellation. The modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from one or several groups of constellations each comprising one or more constellations.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims priority to U.S. application Ser. No. 17/451,112, filed Oct. 15, 2021, which is a continuation of U.S. application Ser. No. 17/119,678, filed Dec. 11, 2020 (now U.S. Pat. No. 11,190,387, issued Nov. 30, 2021), which is a continuation of U.S. application Ser. No. 16/406,746, filed May 8, 2019 (now U.S. Pat. No. 10,868,706, issued Dec. 15, 2020), which is a continuation of U.S. application Ser. No. 16/105,666, filed Aug. 20, 2018 (now U.S. Pat. No. 10,326,635, issued Jun. 18, 2019), which is a continuation of U.S. application Ser. No. 15/699,635, filed Sep. 8, 2017 (now U.S. Pat. No. 10,091,045, issued Oct. 2, 2018), which is a continuation of U.S. application Ser. No. 15/340,672, filed Nov. 1, 2016 (now U.S. Pat. No. 9,813,279, issued Nov. 7, 2017), which is a continuation of U.S. application Ser. No. 14/901,894, filed Dec. 29, 2015 (now abandoned), which is a National Stage of PCT/EP2014/064404, filed Jul. 7, 2014, and claims the benefit of priority to European Patent Application No. 14168129.6, filed May 13, 2014, European Patent Application No. 14153438.8, filed Jan. 31, 2014, European Patent Application No. 13183318.8, filed Sep. 6, 2013, and European Patent Application No. 13175370.9, filed Jul. 5, 2013, the contents of each of which are hereby incorporated by reference in their entirety.

BACKGROUND Field of the Disclosure

The present disclosure relates to a coding and modulation apparatus and method. Further, the present disclosure relates to a transmission apparatus and method. Still further, the present disclosure relates to a computer program and a non-transitory computer-readable recording medium.

Description of Related Art

Modern communications systems typically employ, among other elements, a coding and modulation apparatus (as part of a transmission apparatus) and a decoding and demodulation apparatus (as part of a receiving apparatus). The coding and modulation apparatus is often part of a so called BICM (Bit Interleaved Coded Modulation) apparatus, which generally comprises (at the transmitter side) a serial concatenation of a FEC (Forward Error Correction) encoder, a bit interleaver, and a modulator, which uses spectral efficient modulation such as multilevel PAM (Pulse Amplitude Modulation), PSK (Phase Shift Keying), or QAM (Quadrature Amplitude Modulation). It should be noted that hereinafter, whenever QAM is mentioned it should be understood as a generally term covering PAM, PSK and QAM.

BICM allows for good performance over both non-fading and fading channels due to the use of the interleaver and/or the FEC encoder. It has a reasonable decoding complexity as opposed to multilevel coding (MLC) coding schemes and is thus used frequently in communications systems, such as in all DVB systems, powerline communications (e.g., Homeplug AV, DAB, LTE, WiFi, etc.).

Generally, the coding and modulation capacity, such as the BICM capacity in systems using a BICM apparatus, is considered as a target function, and it is desired to find optimum constellation points such that this capacity is maximized, often subject to a power normalization, i.e., the average power of the constellation points should be normalized to e.g. 1.

The “background” description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventor(s), to the extent it is described in this background section, as well as aspects of the description which may not otherwise qualify as prior art at the time of filing, are neither expressly or impliedly admitted as prior art against the present disclosure.

SUMMARY

It is an object to provide a coding and modulation apparatus and method providing an increased or even maximized capacity, a reduced bit error rate and reception with a reduced SNR (signal-to-noise ratio). It is a further object to provide a corresponding computer program and a non-transitory computer-readable recording medium for implementing said methods.

According to an aspect there is provided a coding and modulation apparatus comprising

-   -   an encoder that encodes input data into cell words, and     -   a modulator that modulates said cell words into constellation         values of a non-uniform constellation,         wherein said modulator is configured to use, based on the total         number M of constellation points of the constellation and the         code rate,

-   i) a non-uniform constellation from a group A or B of constellations     comprising one or more of the following constellations defined by     the constellation position vector u of length v=sqrt(M)/2−1,     -   wherein in one or more constellation position vectors of the         constellations from group B two or more constellation positions         are identical resulting from a condensation of preliminary         constellation positions optimized before,         or

-   ii) a non-uniform constellation from a group C or D of     constellations comprising one or more of the following     constellations, wherein the constellation points are defined by a     constellation position vector w_(0 . . . b-1) with b=M/4, wherein     -   a first quarter of the constellation points x_(0 . . . b-1) are         defined as x_(0 . . . b-1)=w_(0 . . . b-1),     -   a second quarter of the constellation points x_(b . . . 2b-1)         are defined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),     -   a third quarter of the constellation points x_(2b . . . 3b-1)         are defined as x_(2b . . . 3b-1)=−conj(w_(0 . . . b-1)), and     -   a fourth quarter of the constellation points x_(3b . . . 4b-1)         are defined as x_(3b . . . 4b-1)=−w_(0 . . . b1),     -   wherein conj is the complex conjugate,     -   wherein in one or more constellation position vectors of the         constellations from group D two or more constellation positions         are identical resulting from a condensation of preliminary         constellation positions optimized before,         or

-   iii) a non-uniform constellation from a group E of constellations     comprising one or more of the following constellations, wherein the     constellation points are defined by a constellation position vector     w_(0 . . . M-1),     or

-   iv) a non-uniform constellation obtained from a constellation from     anyone of groups A, B, C, D or E through rotation by an angle around     the origin, through inversion of bit labels for all constellation     points, through interchanging of bit positions and/or through     predistortion for the (in particular for all) constellation points,     wherein the constellation position vectors of the different     constellations of the groups A, B, C, D and E of constellations are     defined as follows, wherein the respective values R of the code rate     shall be understood as range of code rates R±1/30.

According to a further aspect there is provided a transmission apparatus comprising

-   -   a coding and modulation apparatus as proposed herein that         encodes and modulates input data into constellation values,     -   a converter that converts said constellation values into one or         more transmission streams to be transmitted, and     -   a transmitter that transmits said one or more transmission         streams.

According to still further aspects corresponding methods, a computer program comprising program means for causing a computer to carry out the steps of the coding and modulation method disclosed herein, when said computer program is carried out on a computer, as well as a non-transitory computer-readable recording medium that stores therein a computer program product, which, when executed by a processor, causes the coding and modulation method disclosed herein to be performed are provided.

Preferred embodiments are defined in the dependent claims. It shall be understood that the claimed methods, the claimed computer program and the claimed computer-readable recording medium have similar and/or identical preferred embodiments as the claimed apparatus and as defined in the dependent claims.

One of the aspects of the disclosure is that the constellation points of the used non-uniform constellations (herein also called NUCs) are not located on a regular grid with equidistant symbols, but rather on optimized locations, dependent on the code rate of forward error correction encoder, like an LDPC or a turbo code or any other known code encoder (generally another forward error correction code encoder, e.g. a BCH encoder, may be provided in addition). Further, the used constellation may be selected (preferably in advance, but generally on the fly in other embodiments) dependent on the desired total number of constellation points of the used constellation (and, in some embodiments, on the channel characteristics).

In the tables various constellations are provided for different values of M and for different code rates. It should be noted that the code rate R indicated in the tables are not to be understood such that a particular constellation is only valid for exactly this code rate, but also for slightly different code rates, i.e. a range of code rates R±1/30. For instance, the indication 6/15 for a code rate (i.e. R=6/15) given in a proposed table means that the respective constellation is valid for a range of code rates 6/15±1/30, i.e. for the range of codes rates from 11/30 to 13/30.

It should also be noted that one or more of the following “invariant transformations” do not affect the properties of the constellations:

1. rotation of all symbols by an arbitrary angle φ,

2. inversion of m-th bit y_m=b∈{0,1} to y_m= b, where the bar indicates inversion,

3. interchanging of bit positions y_k1 and y_k2,

4. reflection on Re{x1}- and/or Im{x1}-axis,

5. predistortion (particularly for constellations of group E).

Thus, the modulator may also use a non-uniform constellation obtained from a constellation from anyone of groups A, B, C, D or E through rotation by an angle around the origin, an inversion of bit labels for all constellation points, an interchanging of bit positions for all constellation points and/or a reflection on the real part and/or imaginary part axis. For instance, if one constellation point has bit labels 0010 for 16-QAM, all first bit labels can be inverted such that this point becomes 1010. Further, constellation obtained through any other trivial manipulation, such as rounding of the constellation points' positions shall generally be covered by the claims. Through one or more of these operations an equivalent mapping to the mapping of the constellations defined in the above mentioned four groups is achieved.

In case of satellite transmission, the modulator might as well transmit different constellation points, obtained by predistortion of proposed constellation points. This predistortion should can act as a countermeasure to the non-linearities of other blocks in the transmission system, in particular the power amplifier. The output of the transmission system however, should correspond to the transmission of the proposed constellations, such that the receiver might assume that these constellations have been transmitted.

It should be noted that to every M-QAM, one can also think of the underlying sqrt(M)-PAM. Further, it should be noted that in other aspects the group of constellations defined in the claims comprises less constellations. e.g. only constellations for non-fading channels, only constellations for fading channels, only constellations for selected values of M, only constellation for M-QAM or sqrt(M)-PAM and/or constellations for less SNR values. In other words, less constellations may be contained in the group of constellations available for selection and subsequent use by the modulator, i.e. the group of constellations available for use by the modulator may comprise one or more of the constellations defined in the claims. Accordingly, the present disclosure is also directed to a coding and modulation apparatus and method that have a smaller group of constellations available for use (as explained above) and/or where less constellations are available for a particular value of M.

A QAM mapping consisting of M constellation points is denoted as M-QAM. These constellations are summarized in group A. If a (uniform or non-uniform) QAM allows separate encoding and decoding of each of its two dimensions (“inphase” and “quadrature phase” in the literature), then this QAM will be called a N²-QAM. This implies that the constellation can be designed by two N-PAM constellations, one for each dimension. N²-QAMs have significantly lower decoding complexity for ML-decoding, as only N constellation points have to be investigated, compared with N² points for the M-QAM, when M=N², but when the two dimensions cannot be separated (as is usually the case for N-PSK, e.g. 8-PSK, where 8 points are located on a unit circle). In addition QAM constellations that are completely defined by a quarter of constellation values of the constellation will be called QQAM, with the other three quarters of constellation values being derived from the first quarter. E.g. normal uniform square QAM constellations (UC) are also QQAM constellations, due to their symmetry. These QQAM constellations are summarized in group C.

However, the constellation points of the QAM constellations according to embodiments considered in this disclosure are not located on a regular grid with equidistant symbols, but rather on optimized locations, dependent on the code rate.

According to the present disclosure an N²-NUC optimization based on N-PAM optimization is considered, combined with a dynamic reduction of the number of constellation points guaranteeing a well defined performance with respect to the performance of the N²-NUC without reduction of the number of constellation points.

Constellation sizes up to 1024-QAM will be considered, where large shaping gains are possible, especially in the high SNR region. By means of a dynamic reduction (also called condensation in the following) of constellation points that are close to each other, the number of constellations points and, thus, the required storage and decoding capacity can be significantly reduced. These constellations are summarized in groups B and D. For example, the 1024-Q-QAM constellation optimized for code rate 6/15 can be reduced from 1024 to 268 constellation points without significant impact on the performance.

It should be noted that the constellation position vector w as defined in the claims directed to a preferred embodiment needs not necessarily contain the constellation points of the first quarter of constellation points of the constellation, but could also contain the constellation points of any of the four quarters (expressed by the definition “of a first quarter” in the claims). Due to the symmetry (which is somewhat provided, but may not be readily visible by the bit labels; the constellation points are generally symmetric with respect to the quadrants) this leads to constellations with a different bit mapping but with identical performance. The constellation position vector w in the tables defined herein should therefore be considered as an example for all four symmetric constellations with different bit mapping but identical performance.

It is to be understood that both the foregoing general description of the disclosure and the following detailed description are exemplary, but are not restrictive, of the disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein:

FIG. 1 shows an embodiment of a coding and modulation apparatus according to the present disclosure,

FIG. 2 shows an embodiment of a transmission apparatus according to the present disclosure,

FIG. 3 shows an embodiment of a communications system according to the present disclosure,

FIG. 4 shows a regular 4-QAM constellation as a simple example for a constellation,

FIG. 5A shows a 8-PAM non-uniform constellation,

FIG. 5B shows a 64-QAM non-uniform constellation,

FIG. 6 shows a constellation for a 64-QAM non-uniform constellation generally defining the constellation points,

FIG. 7 shows a non-uniform 16-QQAM constellation,

FIG. 8 shows a diagram illustrating the performance of non-uniform N²-QAM constellations.

FIG. 9 shows an example for 1D condensing according to an embodiment of the present disclosure,

FIG. 10 shows an example for 2D condensing according to an embodiment of the present disclosure,

FIG. 11A shows a non-uniform 1024-QAM constellation at 27 dB SNR,

FIG. 11B shows a non-uniform 1024-QAM constellation at 8 dB SNR,

FIG. 11C shows an enlargement of the N²-NUC optimized for 8 dB SNR.

FIG. 12A shows a non-uniform 64-QQAM constellation,

FIG. 12B shows a non-uniform 64-QQAM constellation,

FIG. 13 shows a diagram for a constellation from group E,

FIG. 14 shows a diagram illustrating the shaping gain for the above explained non-uniform 8-ary QAM constellations, and

FIGS. 15AA-15CB show examples of LDPC codes for which the disclosed constellations have been optimized.

DESCRIPTION OF THE EMBODIMENTS

Referring now to the drawings, wherein like reference numerals designate identical or corresponding parts throughout the several views, FIG. 1 shows an embodiment of a coding and modulation apparatus 10 according to the present disclosure. It comprises an encoder 11 that encodes input data into cell words, and a modulator 12 that modulates said cell words into constellation values of a non-uniform constellation. Said modulator 12 is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from one of several groups of constellations. Details of those different groups of constellations will be explained in more detail below.

In other embodiments of the coding and modulation apparatus 10 additional elements may be provided, such as a BCH encoder, an LDPC encoder (whose code rate is of interest for selecting and using an optimized constellation by the modulation disclosed herein), a bit interleaver and/or a demultiplexer (for demultiplexing bits of encoded data into the cell words). Some or all of these elements may separate elements or may be part of the encoder 11. For instance, a BICM device as conventionally used in the transmission apparatus of a DVB system may be used as coding and modulation apparatus 10.

FIG. 2 shows an embodiment of a transmission apparatus 20 according to the present disclosure comprising a coding and modulation apparatus 21 (referenced by 10 in FIG. 1 ) as proposed herein that encodes and modulates input data into constellation values, a converter 22 that converts said constellation values into one or more transmission streams to be transmitted, and a transmitter 23 that transmits said one or more transmission streams. In an exemplary embodiment the converter 22 may comprise one or more elements like a time, cell and/or frequency interleaver, a frame builder, an OFDM modulator, etc., as e.g. described in the various standards related to DVB and as may be provided in a coding and modulation apparatus in an upcoming ATSC standard. Other coding and modulation in accordance with other standards, such as the DVB-S2 or subsequent DVB-Sx standards, may not comprises one or more of those elements. The constellations and the constellations values are generally predetermined and e.g. stored in a constellations storage 24 or retrieved from an external source.

In other embodiments of the transmission apparatus 20 additional elements may be provided, such as an input processing unit, a frame building unit and/or an OFDM generation unit as e.g. conventionally used in a transmission apparatus of a DVB system.

FIG. 3 shows an embodiment of a communications system 30 according to the present disclosure comprising one (or more) transmission apparatus 20 (Tx) as shown in FIG. 2 and one or more receiving apparatus 40, 40′ (Rx).

A receiving apparatus 40 generally comprises a receiver 41 that receives one or more transmission streams, a deconverter 42 that deconverts the received one or more transmission streams into constellation values, and a demodulation and decoding apparatus 43 that demodulates and decodes said constellation values into output data. The demodulation and decoding apparatus 43 generally comprises a demodulator 44 for demodulating constellation values of a non-uniform constellation into cell words, and a decoder 45 for decoding cell words into output data words, wherein based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation is selected from the one of several groups of constellations comprising the same predetermined constellations as used in the coding and modulation apparatus 10.

The preferred demodulation and decoding considers soft values as opposed to hard decided values (0 and 1). Soft values represent the continuously distributed received values (possibly after A/D conversion including quantization) by more than two states (as in the case of binary (hard) decision). The reason is that for hard decision, the NUCs are generally not optimal. Nowadays, BICM receivers typically are soft receivers anyway.

Generally, data (e.g. communications data, broadcast data, etc.) shall be transmitted from a transmission apparatus 20 to one or more of said receiving apparatus 40 over a transmission channel 50, 50′. The transmission channel 50, 50′ can be unicast channel, multicast channel, a broadcast channel and may be employed as one-directional or bi-directional channel (i.e. having a return channel from the receiving apparatus to the transmission apparatus).

In an embodiment the modulator 12 is configured to use a non-uniform constellation based on the total number M of constellation points of the constellation, the required signal-to-noise ratio SNR for error free decoding in dB and the channel characteristics. In broadcasting applications the constellation is generally not selected dependent on the SNR in the receiver, but dependent on the SNR that is required for error free decoding with a used channel code (if a code is used, for example LDPC codes in case of DVB 2^(nd) generation transmission systems) for an expected channel characteristic, e.g., static reception or multipath fading.

For the broadcaster there may be a trade-off: using small constellation sizes M and/or low code rates R allow robust transmission (reception also at low SNR), but the throughput of the system scales with log₂(M)*R. For instance, a 16-QAM with code rate R=3/4 can transmit three information bits per coded QAM symbol. This results in relatively small spectral efficiency. On the other hand, high (spectral) efficiency requires large SNR. Thus, the constellations should allow decreasing the SNR required for successful decoding, while leaving the efficiency constant. This is the so called “shaping gain” of the optimized constellations.

The total number M of constellation points is generally selected according to the desired payload throughput jointly with the code rate of the FEC encoder. The SNR for error free decoding for typical channel characteristic is generally known, e.g. by simulation. In broadcasting the channel characteristics of the receivers are not known, i.e. a compromise is selected. For instance, in broadcasting for each code rate of the FEC encoder one non-uniform constellation is selected, optimized for an SNR that is a compromise for all channel characteristics.

The transmitter generally targets a certain scenario. For instance, a broadcast transmission over cable or satellite considers the channel to be just a non-fading AWGN (appropriate channel model), while a terrestrial broadcaster typically considers the channel to be a fading channel, e.g. with Rayleigh distribution, as several echoes are usually received. Preferably, the proposed constellations consider the most relevant channel characteristics.

In another embodiment the modulator 12 is configured to adaptively select a non-uniform constellation based on the total number M of constellation points of the constellation, the signal-to-noise ratio SNR in dB and the channel characteristics, wherein said signal-to-noise ratio SNR in dB and channel characteristics are received from a receiving device 40 to which data shall be transmitted. Such an adaptive selection of the constellation is generally only possible with a return channel in unicast environments. A non-uniform constellation may be adapted e.g. in time and/or frequency domain, e.g. for different OFDM subcarriers.

The channel characteristics describe the statistical properties of the channel, e.g., the extent of the multipath propagation of the transmission channel between transmitter and receiver. If the channel is characterized by no multipath propagation, corresponding to the AWGN channel, the required SNR for error free decoding is relatively low, i.e. the NUC has to be selected accordingly for optimum performance. If the transmission channel is characterized by strong multipath propagation, the required SNR for error free reception is larger compared to a channel without multipath propagation, i.e. a NUC optimized for higher SNR has to be used. Further, the NUCs should be optimized taking the fading characteristics into account, as will be discussed below.

As mentioned above, the number M of the constellation points of the constellations is selected according to the desired payload throughput. Larger values of M allow for higher data throughput, but require a larger SNR for error free reception. This is further influenced by the code rate of the FEC encoder, if any FEC encoder is used.

Another explanation (which is closely related to the optimization task) is that the performance of the constellation in combination with a forward error correction code (e.g. LDPC and/or BCH code) having a certain code rate shall be optimized. Thus, for various codes/code rates optimized constellations are proposed for different values of M. Another optimization target is the capacity. For an expected SNR, say 15 dB of SNR should be guaranteed, M is chosen, for which the respective optimized NUC yields the largest capacity. As a general rule it holds that for low SNR a low value of M should be selected and vice versa. But from a theoretical point of view, it turns out that high M is generally optimum, e.g., choosing M=4096 or M=1024 is preferred, because even for low SNR, the optimized NUC will “look (almost) like” a constellation with effectively smaller M, as several points will overlap. However, modulation and demodulation complexity increase with increasing M, so a tradeoff is considered. Another aim is to reduce the BER (bit error rate) and/or the FER (frame error rate) and/or to achieve the same BER and/or the FER at a lower SNR compared to a situation where a “normal” (not optimized) constellation is used.

A simple example of a constellation is shown in FIG. 4 . Here a regular 4-QAM consisting of constellation points (e^(jπ/4), e^(j7π/4), e^(3π/4), e^(j5π/4)) is depicted. The average symbol power is 1 (all symbols are located on unit circle here). The above symbol vector (e^(jπ/4), e^(j7π/4), e^(3π/4), e^(j5π/4)) is to be understood such that the first entry (e^(jπ/4)) belongs to the bit vector 00, the second entry (e^(j7π/4)) to 01 and so on, i.e. the entries belong to bit vectors with increasing values, where the first bit position is the most significant bit (MSB) and the last one the least significant bit (LSB). This 4-QAM is a particular case of an N²-QAM, with N=2. Note that this definition (of being an N² QAM) does not only require N² being a square number (N²=2²), but also that the constellation is symmetrical and can be described by two independent N-PAM constellations, here a 2-PAM: the in-phase component (real-part of the complex symbols) is a 2-PAM with symbol vector (1/sqrt(2), −1/sqrt(2)) and describes the 1^(st) bit of the 4-QAM, whereas the quadrature-phase component (imaginary-part of the complex symbols) is the same 2-PAM, this time describing the 2^(nd) bit of the 4-QAM. Note further that the decomposition of the N²-QAM into two N-PAMs is only possible if the bit labelling is according to binary reflected Gray mapping, which is typically applied (e.g. in DVB-systems).

The above example can be extended to higher order N²-QAMs, with N>2. Then the underlying N-PAM describes for one component the 1^(st), 3^(rd), 5^(th) and so on bit label, while for the other component it describes the 2^(nd), 4^(th), 6^(th) and so on label.

All constellations preferably fulfil power constraint, i.e

${\mathcal{P}_{x} = {{E_{x_{l}}\left\lbrack {❘x_{l}❘}^{2} \right\rbrack} = {{\frac{1}{M}{\sum_{l = 0}^{M - 1}{❘x_{l}❘}^{2}}}\overset{!}{=}1}}},$ where E[.] is the expectation operator, and x₁ is a particular symbol of the set of all M constellation symbols.

N²-NUCs have been optimized as one embodiment with N² being 16, 64, 256, 1024 (1 k). This means that these constellations are optimized to allow minimum BER/FER for a given FEC code rate. The restriction on these constellations is that they should be able to be split into two separate one-dimensional PAM constellations, allowing low complexity mapping at the transmitter and demapping at the receiver.

As an example, a M=64 NUC described here yield the following values (an example from the tables could yield these three numbers, then there is the 1 at the beginning (normalization due to power constraint) and so on):

-   -   2.2794 4.6229 7.5291.         This means that the positive constellation values are     -   1 2.2794 4.6229 7.5291         (the 1 was redundant, due to the power normalization, which will         be applied in the end). The underlying 1-dim. 8-PAM NUC is thus         described by the symbol vector     -   (1.6405 1.0073 0.2179 0.4967 −1.6405 −1.0073 −0.2179 −0.4967),         where the values are already normalized to unit average power.

As described before, the first entry (1.6405) corresponds to the bit label 000, the next one (1.0073) to 001 and so on. The 2-dim. 64-NUC is then obtained by symmetry, where both in-phase and quadrature-phase component of the NUC are based on the 8-PAM NUC.

FIG. 5A depicts 8-PAM NUC. FIG. 5B depicts 64-QAM NUC. The bit labels are given in integer numbers (000→0, 001→1, 010→2 and so on).

The creation of the 2-dim. NUC based on the optimized degrees of freedom will be explained in more detail below.

Since the performance of NUCs depends on the SNR value they are optimized for, a thorough selection is preferably carried out depending on the (FEC) code rate to achieve optimum performance. If the channel characteristics are known, the required SNR value for FEC convergence can be determined by simulation. Then the NUC that has been optimized for this SNR value is chosen for best performance. If the SNR at the receiver is lower than this SNR decoding threshold, the constellation is not optimal. However, this is no drawback, since the capacity is too low for successful decoding anyhow. On the other hand if the SNR at the receiver is clearly higher than the decoding threshold, a sufficient amount of capacity for successful decoding is available, even though the NUC is suboptimal for this SNR range. Therefore, the NUC needs to be optimized for the SNR value at the waterfall region (i.e., decoding threshold for (quasi-) error free decoding) of the FEC. As the SNR value of the waterfall region depends on the code rate of the FEC, a different NUC is selected for each code rate.

The SNR value for (quasi-) error free decoding also depends on the channel characteristics of the receiver. For instance the required SNR for error free decoding of the DVB-T2 LDPC code in the AWGN channel is 0.8 dB, whereas 2.5 dB are required in the Rayleigh P1 multipath channel. The selected NUC for each code rate is thus not optimal in all channel environments and a tradeoff is necessary in a broadcasting environment that suits all (or most) users in the network. In a point-to-point network with return channel, the optimal NUC may be selected based on the measured channel characteristics in the receiver.

In the following some more explanation is provided regarding the definition of the non-uniform QAM constellations. Each input cell word (y_(0,q) . . . y_(m-1,q)) (i.e. provided to the modulator) shall be modulated using a non-uniform QAM constellation to give a constellation point z_(q) prior to normalization, where m corresponds to the number of bits per QAM symbol m=log₂(M). It should be noted that the parameter q used here for discrete time or subcarrier index corresponds to the parameter k as used in the above. The exact values of the real and imaginary components Re(z_(q)) and Im(z_(q)) for each combination of the relevant input bits y_(0 . . . m-1,q) are given in the following tables for the various constellation sizes depending on the NUC position vector u_(1 . . . v), which defines the constellation point position of the non-uniform constellation. The length of the NUC position vector u is defined by

$v = {\frac{\sqrt{M}}{2} - 1.}$

In one example, the corresponding constellation point z_(q) for a 64-QAM NUC defined by the NUC position vector (u_(1 . . . 3))=(2,5,6) and the input cell word (y_(0,q) . . . y_(m-1,q))=(100111) is Re(z_(q))=−u₂=−5 and Im(z_(q))=u₁=2. The complete constellation for this NUC position vector is shown in FIG. 6 with exemplary input cell words marked at the corresponding constellation points.

The resulting constellation mapping (also called labeling) for the non-uniform constellations follows a binary reflected Gray-Mapping (labeling), i.e. neighboring constellation points differ in only one bit. The power of the constellation points z_(q) is normalized such that the expectation value of the normalized constellation point f_(q) equals 1, i.e. E(|f_(q)|²)=1. For example, the normalized constellation value f_(q) of a uniform 16-QAM constellation results by

$f_{q} = {\frac{z_{q}}{\sqrt{10}}.}$

The following tables define the constellation position vectors (prior to power normalization) as well as the bit labelling of the data cell words to the constellation points.

Constellation mapping for real part of 16-QAM y_(0, q) 1 1 0 0 y_(2, q) 0 1 1 0 Re(z_(q)) −3 −1 1 3 Uniform −u₁ −1 1 u₁ NUC

Constellation mapping for imaginary part of 16-QAM y_(1, q) 1 1 0 0 y_(3, q) 0 1 1 0 Im(z_(q)) −3 −1 1 3 Uniform −u₁ −1 1 u₁ NUC

Constellation mapping for real part of 64-QAM y_(0,q) 1 1 1 1 0 0 0 0 y_(2,q) 0 0 1 1 1 1 0 0 y_(4,q) 0 1 1 0 0 1 1 0 Re(z_(q)) −7 −5 −3 −1 1 3 5 7 Uniform −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ NUC

Constellation mapping for imaginary part of 64-QAM y_(1,q) 1 1 1 1 0 0 0 0 y_(3,q) 0 0 1 1 1 1 0 0 y_(5,q) 0 1 1 0 0 1 1 0 Im(z_(q)) −7 −5 −3 −1 1 3 5 7 Uniform −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ NUC

Constellation mapping for real part of 256-QAM y_(0,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(2,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(4,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(6,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −15 −13 −11 −9 −7 −5 −3 −1 1 3 5 7 9 11 13 15 Uniform −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ NUC

Constellation mapping for real part of 256-QAM y_(1,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(3,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(5,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(7,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) −15 −13 −11 −9 −7 −5 −3 −1 1 3 5 7 9 11 13 15 Uniform −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ NUC

Constellation mapping for real part of 1024-QAM Y_(0,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(2,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(4,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(6,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(8,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −31 −29 −27 −25 −23 −21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁ −u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC Y_(0,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(2,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(4,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(6,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(8,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅ NUC

Constellation mapping for imaginary part of 1024-QAM y_(1,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(3,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(5,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(7,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(9,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) −31 −29 −27 −25 −23 −21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁ −u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC y_(1,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(3,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(5,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(7,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(9,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅ NUC

Constellation mapping for real part of 4096-QAM Y_(0,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(2,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(4,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −63 −61 −59 −57 −55 −53 −51 −49 −47 −45 −43 −41 −39 −37 −35 −33 Uniform −u₃₁ −u₃₀ −u₂₉ −u₂₈ −u₂₇ −u₂₆ −u₂₅ −u₂₄ −u₂₃ −u₂₂ −u₂₁ −u₂₀ −u₁₉ −u₁₈ −u₁₇ −u₁₆ NUC Y_(0,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(2,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(4,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) −31 −29 −27 −25 −23 −21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u₁₅ −u₁₄ −u₁₃ −u₁₂ −u₁₁ −u₁₀ −u₉ −u₈ −u₇ −u₆ −u₅ −u₄ −u₃ −u₂ −u₁ −1 NUC Y_(0,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(2,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(4,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Uniform 1 u₁ u₂ u₃ u₄ u₅ u₆ u₇ u₈ u₉ u₁₀ u₁₁ u₁₂ u₁₃ u₁₄ u₁₅ NUC Y_(0,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(2,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(4,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(6,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(8,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(10,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Re(z_(q)) 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 Uniform u₁₆ u₁₇ u₁₈ u₁₉ u₂₀ u₂₁ u₂₂ u₂₃ u₂₄ u₂₅ u₂₆ u₂₇ u₂₈ u₂₉ u₃₀ u₃₁ NUC

Constellation mapping for imaginary part of 4096−QAM y_(1,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(3,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(5,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(7,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(9,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(11,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) −63 −61 −59 −57 −55 −53 −51 −49 −47 −45 −43 −41 −39 −37 −35 −33 Uniform −u31 −u30 −u23 −u28 −u27 −u26 −u25 −u24 −u23 −u22 −u21 −u20 −u19 −u18 −u17 −u16 NUC y_(1,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(3,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(5,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(7,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(9,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(11,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) −31 −29 −27 −25 −23 −21 −19 −17 −15 −13 −11 −9 −7 −5 −3 −1 Uniform −u15 −u14 −u13 −u12 −u11 −u10 −u9 −u8 −u7 −u6 −u5 −u4 −u3 −u2 −u1 −1 NUC y_(1,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(3,q) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 y_(5,q) 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 y_(7,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(9,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(11,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 Uniform 1 u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u15 NUC y_(1,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(3,q) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y_(5,q) 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 y_(7,q) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 y_(9,q) 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 y_(11,q) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 Im(z_(q)) 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 Uniform u16 u17 u18 u19 u20 u21 u22 u23 u24 u25 u26 u27 u28 u29 u30 u31 NUC

In an embodiment the modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from a group A of constellations comprising one or more of the following constellations defined by the constellation position vector u of length v=sqrt(M)/2−1.

The following non-uniform constellations comprised in group A are proposed:

A) M-QAM non-uniform constellations of group A:

A1) 16-QAM NUC

u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15 u1 3.1169 3.1973 3.2334 3.2473 3.2436 3.2312 3.2149 3.1976 A2) 64-AM NUC

u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15 u1 1.5409 2.1208 2.3945 2.6067 2.8505 2.912 2.9751 3.0032 u2 3.5826 4.3237 4.6577 4.9074 5.1674 5.2201 5.2491 5.2489 u3 5.5069 6.8108 7.3475 7.7177 8.0398 8.068 8.0217 7.9528 A3) 256-QAM NUC

u/code rate 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15 u1 0.9918 0.9989 1.1155 1.3963 2.2282 2.6619 2.8437 2.9176 u2 2.2615 2.6086 2.8419 3.1795 4.1541 4.664 4.8758 4.956 u3 2.2873 2.7307 3.2659 3.9675 5.676 6.5386 6.8857 7.0096 u4 4.2761 4.6692 5.1383 5.9281 7.9072 8.8521 9.1906 9.282 u5 4.6871 5.3576 6.3082 7.4353 10.0292 11.2248 11.6157 11.6881 u6 6.5483 7.3828 8.4196 9.7825 12.8864 14.2018 14.5326 14.508 u7 8.6107 9.7612 11.0879 12.7927 16.5632 17.9894 18.1926 17.9984 A4) 1024-QAM NUC

u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15 u1 0.9997 1.0003 0.9994 1.0005 1.0023 1.0772 2.0789 2.5983 2.8638 u2 0.9916 1.0149 1.2742 2.0897 2.5667 2.8011 3.9147 4.5193 4.8422 u3 0.9911 1.0158 1.2749 2.0888 2.5683 2.9634 5.0664 6.1649 6.7392 u4 2.4349 2.6848 3.0323 3.9945 4.5468 4.8127 7.0579 8.2107 8.7961 u5 2.4346 2.6903 3.0371 3.9931 4.5636 5.1864 8.3596 9.9594 10.7659 u6 2.486 2.882 3.6813 5.3843 6.2876 6.7838 10.2901 12.0321 12.8844 u7 2.4864 2.8747 3.6718 5.3894 6.4073 7.5029 11.834 13.9574 14.98 u8 4.4576 4.7815 5.5854 7.5206 8.4282 9.238 14.0092 16.2598 17.2736 u9 4.4646 4.7619 5.5804 7.6013 8.8692 10.32 15.8419 18.4269 19.5552 u10 4.9706 5.5779 6.8559 9.3371 10.6112 12.0115 18.1472 20.9273 22.0472 u11 4.9552 5.6434 7.0475 9.8429 11.6946 13.5356 20.4243 23.4863 24.6335 u12 6.7222 7.3854 8.8436 11.9255 13.7334 15.6099 23.2381 26.4823 27.5337 u13 7.0327 7.8797 9.7042 13.3962 15.6274 17.7524 26.2322 29.7085 30.6651 u14 8.53S2 9.635 11.727 15.8981 18.2933 20.5256 29.973 33.6247 34.3579 u15 10.4411 11.7874 14.2894 19.1591 21.7769 24.1254 34.7629 38.5854 38.9603 A5) 4096-QAM NUC

u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/15 12/15 13/15 u1 1.0004 0.9998 0.9988 0.9999 0.9999 1.0009 1.0043 2.031 2.7135 u2 1.0009 0.9984 0.9996 1.0004 1.0364 2.272 2.7379 3.8448 4.6426 u3 1.0008 0.9983 0.998 1.0013 1.037 2.2732 2.746 4.8763 6.3617 u4 0.9953 1.2356 2.0374 2.5341 2.7604 4.1332 4.6828 6.8402 8.3598 u5 0.9956 1.2354 2.0374 2.5347 2.7604 4.1327 4.7079 7.912 10.0999 u6 0.9956 1.2372 2.0374 2.5328 2.8536 5.4651 6.4365 9.7373 12.0627 u7 0.9957 1.2366 2.0366 2.5362 2.8534 5.4655 6.5065 10.8859 13.8413 u8 2.5769 2.9864 3.93 4.5077 4.7342 7.4777 8.4445 12.8891 15.9032 u9 2.5774 2.9867 3.931 4.5089 4.7345 7.478 8.6065 14.1091 17.7263 u10 2.5794 2.9906 3.9297 4.5202 5.0028 8.9382 10.254 15.9704 19.7625 u11 2.5793 2.9912 3.9294 4.5204 5.0025 8.9384 10.5627 17.2797 21.6431 u12 2.6756 3.5858 5.277 6.2318 6.6184 10.9166 12.3025 19.2948 23.7857 u13 2.6764 3.5859 5.2757 6.2314 6.6189 10.9185 12.8281 20.7185 25.7402 u14 2.6747 3.577 5.2792 6.3219 7.215 12.5855 14.3539 22.6727 27.9051 u15 2.6746 3.5775 5.2795 6.3212 7.2149 12.5908 15.1126 24.2439 29.9595 u16 4.6252 5.474 7.4011 8.3718 8.9581 14.7944 16.7817 26.3916 32.2702 u17 4.6243 5.4757 7.3997 8.372 8.9585 14.8184 17.7277 28.1062 34.4336 u18 4.6188 5.4675 7.4646 8.7429 9.94 16.6805 19.2664 30.2335 36.7993 u19 4.62 5.4681 7.4665 8.7398 9.9394 16.8025 20.3968 32.1242 39.1019 u20 5.246 6.7028 9.1828 10.5001 11.5978 18.9131 22.1002 34.4644 41.6513 u21 5.2441 6.7104 9.1919 10.4999 11.6041 19.325 23.4306 36.5679 44.1297 u22 5.2556 6.8816 9.641 11.4966 12.9965 21.168 25.1704 39.027 46.8305 u23 5.2584 6.8738 9.6278 11.5018 13.088 22.0945 26.7556 41.4123 49.5435 u24 7.0279 8.6613 11.7039 13.483 14.878 24.1425 28.7536 44.2002 52.5353 u25 7.0459 8.6344 11.6913 13.5672 15.2586 25.575 30.6214 46.9287 55.5551 u26 7.4305 9.4101 12.9619 15.1187 16.8096 27.6541 32.829 50.0315 58.8707 u27 7.3941 9.5027 13.2128 15.6519 17.8237 29.6567 35.1146 53.2678 62.3471 u28 8.8516 11.1654 15.161 17.6098 19.6965 32.2679 37.8272 57.0085 66.2558 u29 9.1461 11.7322 16.204 19.1046 21.4926 35.0526 40.7685 61.0847 70.5087 u30 10.4828 13.5243 18.4804 21.5413 23.9997 38.6023 44.3725 65.9903 75.5397 u31 12.3176 15.7967 21.4433 24.7641 27.2995 43.2007 48.9596 72.1993 81.8379

In the following the Q-NUC optimization will be described, i.e. the optimization of a 2-dimensional constellation that is derived from a single quadrant. The above described optimization of a N²-QAM requires the optimization of sqrt(M)/2−1 degrees of freedom. Since the optimization of a 2-dimensional QAM constellation has 2*M degrees of freedom (real and imaginary part of each constellation point) the optimization is significantly more time consuming. Since the optimum 2D-constellations for the 16-QAM case are symmetric with respect to the different quadrants of the constellations, the following simplifications can be applied to describe these constellations: Only a first quarter of the total number of constellation points of a constellation (e.g. the first quadrant of the constellation) is described, reducing the number of table entries from 2*M to M/2. From the first quarter the remaining quarters can be derived, leading to a so called QQAM constellation. However, it shall be ensured that the properties of the bit labeling of the constellation points are retained. For instance, if the first quadrant is Gray-Mapped, offering a Hamming distance of 1 of the bit labels belonging to adjacent constellation points, the same must be ensured for the remaining quadrants of the QQAM constellation.

To uniquely define a 16-QQAM only 8 real values are required, corresponding to 4 complex values representing the constellation points of the first quadrant. Based on the QQAM approach 16-QQAM, 32-QQAM, 64QQAM, 128-QQAM, 256-QQAM and 1024-QQAM constellations have been optimized, clearly outperforming the N²-QAM constellations. The presented QQAM optimization approach can be used for any channel condition, e.g. for the AWGN channel as well as for fading channels.

For other systems like a satellite communication system in accordance with the DVB-S2 or Sx standard, constellations for M=8 constellation points have been optimized. These constellations cannot be described by only a quarter of the constellations points. Rather all 8 complex values are explicitly described.

In an embodiment the modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from a group C or D of constellations comprising one or more of the following constellations, wherein the constellation points are defined by a constellation position vector w_(0 . . . n-1) with b=M/4, wherein

a first quarter of the constellation points x_(0 . . . b-1) are defined as x_(0 . . . b-1)=w_(0 . . . b-1),

a second quarter of the constellation points x_(b . . . 2b-1) are defined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),

a third quarter of the constellation points x_(2b . . . 3b-1) are defined as x_(2b . . . 3b-1)=−conj(w_(0 . . . b-1)), and

a fourth quarter of the constellation points x_(3b . . . 4b-1) are defined as x_(3b . . . 4b-1)=−w_(0 . . . b-1),

wherein conj is the complex conjugate.

The following non-uniform constellations comprised in group C are proposed (i=srqt(−1) is the imaginary unit):

C) M-QAM non-uniform constellations of group C:

C1) 16-QQAM NUC

w/code rate 5/15 6/15 7/15 8/15 9/15 w0 1.1928 + 0.5309i 0.2663 + 0.4530i 0.2626 + 0.4634i 0.2575 + 0.4819i 0.2474 + 0.5061i w1 0.5309 + 1.1928i 0.4530 + 0.2663i 0.4634 + 0.2626i 0.4819 + 0.2575i 0.5061 + 0.2474i w2 0.4633 + 0.2842i 0.5115 + 1.2092i 0.5023 + 1.2100i 0.4951 + 1.2068i 0.4909 + 1.2007i w3 0.2842 + 0.4633i 1.2092 + 0.5115i 1.2100 + 0.5023i 1.2068 + 0.4951i 1.2007 + 0.4909i w/code rate 10/15 11/15 12/15 13/15 w0 0.6186 + 0.2544i 0.2881 + 0.9576i −0.2949 + 0.2949i −0.2999 + 0.2999i w1 0.2213 + 0.4416i 0.2881 + 0.2881i −0.9555 + 0.2949i −0.9540 + 0.2999i w2 1.2080 + 0.5377i 0.9576 + 0.9576i −0.2949 + 0.9555i −0.2999 + 0.9540i w3 0.4487 + 1.1657i 0.9576 + 0.2881i −0.9555 + 0.9555i −0.9540 + 0.9540i C2) 64-QQAM NUC

w/code rate 5/15 5/15 7/15 3/15 9/15 w0 1.0257 + 0.5960i 3.5656 + 0.9499i 0.2925 + 1.4892i 3.2920 + 1.4827i 0.2978 + 1.4669i w1 1.2181 + 0.7476i 0.2750 + 1.0676i 0.8449 + 1.2622i 3.8411 + 1.2563i 3.8421 + 1.2355i w2 1.1509 + 0.3069i 3.8202 + 1.2651i 0.2351 + 1.0196i 3.2174 + 1.0211i 3.2135 + 1.0389i w3 1.3S8S + 0.3325i 3.3011 + 1.4529i 0.5555 + 0.8926i 3.5702 + 0.8798i 0.6055 + 0.8654i w4 3.5961 + 1.0257i 0.9500 + 0.5641i 1.4892 + 0.2925i 1.4827 + 0.2920i 1.4685 + 0.2859i w5 3.7476 + 1.2181i 1.0666 + 0.2744i 1.2622 + 0.8449i 1.2563 + 0.8410i 1.2516 + 0.8201i w6 3.3069 + 1.1510i 1.2657 + 0.8178i 1.0196 + 0.2351i 1.0211 + 0.2174i 1.0279 + 0.1981i w7 3.3325 + 1.3888i 1.4521 + 0.3005i 0.8926 + 0.5555i 3.8798 + 0.5702i 3.8857 + 0.5642i w8 3.4969 + 0.2870i 3.3004 + 0.5417i 0.1558 + 0.3029i 9.1475 + 0.3040i 0.1338 + 0.3767i w9 3.4161 + 0.2486i 0.2430 + 0.5607i 0.1712 + 0.3021i 3.1691 + 0.3028i 3.1752 + 0.3563i w10 3.4969 + 0.2500i 3.2094 + 0.3579i 0.2075 + 0.6586i 3.1871 + 0.6855i 3.1756 + 0.7261i w11 3.4084 + 0.2266i 3.1946 + 0.3566i 0.3354 + 0.6030i 0.3563 + 0.6126i 3.4023 + 0.6180i w12 0.2870 + 0.4969i 3.5410 + 0.3002i 0.3029 + 0.1558i 3.3040 + 0.1475i 3.2713 + 0.1337i w13 3.2486 + 0.4162i 3.5596 + 0.2431i 0.3021 + 0.1712i 3.3028 + 0.1691i 3.2748 + 0.1572i w14 3.2500 + 0.4969i 3.3569 + 0.2094i 0.6586 + 0.2075i 0.6855 + 0.1871i 3.6840 + 0.1578i w15 3.2266 + 0.4054i 3.3553 + 0.1948i 0.6030 + 0.3354i 3.6126 + 0.3563i 3.6145 + 0.3556i w/code rate 10/15 11/15 12/15 13/15 w0 0.2878 + 1.4388i −0.4661 + 0.9856i 1.0519 − 0.5188i 1.0854 − 0.5394i w1 0.8133 + 1.2150i −0.4329 + 0.6825i 0.7146 − 0.4532i 0.7353 − 0.4623i w2 0.2219 + 1.0386i −0.1534 + 1.0366i 1.0500 − 0.1642i 1.0474 − 0.1695i w3 0.6145 + 0.8494i −0.1340 + 0.6796i 0.7170 − 0.1473i 0.7243 − 0.1504i w4 1.4656 + 0.2931i −0.8178 + 1.1580i 1.0952 − 0.9115i 1.0693 − 0.9408i w5 1.2278 + 0.8230i −0.8132 + 0.6913i 0.6868 − 0.8108i 0.7092 − 0.8073i w6 1.0649 + 0.2069i −0.2702 + 1.4529i 1.4480 − 0.2403i 1.4261 − 0.2216i w7 0.8971 + 0.5677i −1.2125 + 0.7097i 0.6406 − 1.1995i 0.6106 − 1.1783i w8 0.1177 + 0.4119i −0.4145 + 0.1264i 3.1325 − 0.3998i 0.1392 − 0.4078i w9 0.2516 + 0.3998i −0.4179 + 0.3948i 0.4122 − 0.4120i 0.4262 − 0.4205i w10 0.1559 + 0.7442i −0.1353 + 0.1272i 0.1374 − 0.1295i 0.1407 − 0.1336i w11 0.4328 + 0.5954i −0.1359 + 0.3877i 0.4185 − 0.1357i 0.4265 − 0.1388i w12 0.1678 + 0.1166i −0.7330 + 0.1416i 0.1369 − 0.7073i 0.1388 − 0.7057i w13 0.3325 + 0.1582i −0.7177 + 0.4018i 0.4044 − 0.7057i 0.4197 − 0.7206i w14 0.7408 + 0.1355i −1.0718 + 0.1686i 0.1677 − 1.0405i 0.1682 − 1.0316i w15 0.6200 + 0.3227i −1.4375 + 0.2732i 0.2402 − 1.4087i 0.2287 − 1.3914i C3) 256-QQAM NUC

w/code rate 5/15 6/15 7/15 8/15 9/15 w0 −0.3495 + 1.7549i 0.6800 + 1.6926i 0.7280 + 1.63841 1.0804 + 1.3788i 1.6350 + 0.1593i w1 −0.2804 + 1.4293i 0.3911 + 1.3645i 0.4787 + 1.3492i 1.0487 + 0.9862i 1.5776 + 0.4735i w2 −0.2804 + 1.4293i 0.2191 + 1.7524i 0.2417 + 1.7872i 1.6464 + 0.7428i 1.3225 + 0.1320i w3 −0.2635 + 1.3614i 0.2274 + 1.4208i 0.1966 + 1.4478i 1.8245 + 0.9414i 1.2742 + 0.3922i w4 −0.9918 + 1.4880i 0.6678 + 1.2487i 0.9185 + 1.2490i 0.7198 + 1.2427i 1.2901 + 1.0495i w5 −0.8109 + 1.2116i 0.7275 + 1.1667i 0.7448 + 1.1524i 0.8106 + 1.0040i 1.4625 + 0.7740i w6 −0.8109 + 1.2116i 0.8747 + 1.0470i 0.9536 + 0.9516i 0.5595 + 1.0317i 1.0382 + 0.8623i w7 −0.7744 + 1.1516i 0.7930 + 1.0406i 0.8912 + 0.9461i 0.6118 + 0.9722i 1.1794 + 0.6376i w8 −0.2340 + 0.9857i 0.2098 + 0.9768i 0.2553 + 0.9993i 1.6768 + 0.2002i 0.9430 + 0.1100i w9 −0.2349 + 1.0112i 0.2241 + 1.0454i 0.2988 + 1.0689i 0.9997 + 0.6844i 0.9069 + 0.2829i w10 −0.2349 + 1.0112i 0.1858 + 0.9878i 0.1656 + 1.0288i 1.4212 + 0.4769i 1.0854 + 0.1139i w11 −0.2356 + 1.0372i 0.1901 + 1.0659i 0.1779 + 1.1140i 1.1479 + 0.6312i 1.0441 + 0.3296i w12 −0.5320 + 0.8680i 0.5547 + 0.8312i 0.5802 + 0.8040i 0.6079 + 0.6566i 0.7273 + 0.6160i w13 −0.5496 + 0.8870i 0.5479 + 0.8651i 0.5788 + 0.8534i 0.7284 + 0.6957i 0.8177 + 0.4841i w14 −0.5496 + 0.8870i 0.6073 + 0.8182i 0.6616 + 0.7612i 0.5724 + 0.7031i 0.8504 + 0.7217i w15 −0.5674 + 0.9058i 0.5955 + 0.8420i 0.6574 + 0.7871i 0.6302 + 0.7259i 0.9638 + 0.5407i w16 −1.7549 + 0.3495i 1.4070 + 0.1790i 1.4079 + 0.1358i 0.1457 + 1.4010i 0.1658 + 1.6747i w17 −1.4293 + 0.2804i 1.7227 + 0.2900i 1.7492 + 0.2856i 0.1866 + 1.7346i 0.4907 + 1.6084i w13 −1.4293 + 0.2804i 1.3246 + 0.2562i 1.3108 + 0.2733i 0.1174 + 1.1035i 0.1322 + 1.3631i w19 −1.3614 + 0.2635i 1.3636 + 0.3654i 1.3393 + 0.4031i 0.1095 + 1.0132i 0.3929 + 1.3102i w20 −1.4880 + 0.991Si 1.3708 + 1.2834i 1.3733 + 1.2596i 0.4357 + 1.3636i 1.0646 + 1.2876i w21 −1.2116 + 0.8109i 1.6701 + 0.8403i 1.6601 + 0.8198i 0.5853 + 1.6820i 0.7949 + 1.4772i w22 −1.2116 + 0.8109i 1.1614 + 0.7909i 1.1559 + 0.7249i 0.3439 + 1.0689i 0.8555 + 1.0542i w23 −1.1516 + 0.7744i 1.2241 + 0.7367i 1.2163 + 0.6897i 0.3234 + 0.9962i 0.6363 + 1.2064i w24 −0.9857 + 0.2340i 0.9769 + 0.1863i 0.9601 + 0.1547i 0.1092 + 0.6174i 0.1088 + 0.9530i w25 −1.0112 + 0.2349i 0.9452 + 0.2057i 0.9220 + 0.1683i 0.1074 + 0.6307i 0.2464 + 0.9270i w26 −1.0112 + 0.2349i 1.0100 + 0.2182i 1.0004 + 0.1894i 0.1109 + 0.6996i 0.1124 + 1.1327i w27 −1.0372 + 0.2356i 0.9795 + 0.2417i 0.9581 + 0.2045i 0.1076 + 0.7345i 0.3160 + 1.0913i w28 −0.8680 + 0.5320i 0.8241 + 0.4856i 0.8067 + 0.4374i 0.3291 + 0.6264i 0.5707 + 0.7662i w29 −0.8870 + 0.5496i 0.8232 + 0.4837i 0.8071 + 0.4310i 0.3126 + 0.6373i 0.4490 + 0.8461i w30 −0.8870 + 0.5496i 0.8799 + 0.5391i 0.8603 + 0.4941i 0.3392 + 0.6999i 0.6961 + 0.8850i w31 −0.9058 + 0.5674i 0.8796 + 0.5356i 0.8598 + 0.4876i 0.3202 + 0.7282i 0.5229 + 1.0037i w32 −0.1570 + 0.3085i 0.1376 + 0.3342i 0.1222 + 0.4300i 0.9652 + 0.1066i 0.3237 + 0.0849i w33 −0.1564 + 0.3061i 0.1383 + 0.3292i 0.1222 + 0.4272i 0.9075 + 0.1666i 0.3228 + 0.0867i w34 −0.1564 + 0.3061i 0.1363 + 0.3322i 0.1213 + 0.4297i 0.9724 + 0.1171i 0.4582 + 0.1123i w35 −0.1557 + 0.3037i 0.1370 + 0.3273i 0.1213 + 0.4269i 0.9186 + 0.1752i 0.4545 + 0.1251i w36 −0.1726 + 0.3073i 0.1655 + 0.3265i 0.1716 + 0.3987i 0.6342 + 0.1372i 0.2844 + 0.1296i w37 −0.1716 + 0.3050i 0.1656 + 0.3227i 0.1708 + 0.3972i 0.6550 + 0.1495i 0.2853 + 0.1309i w38 −0.1716 + 0.3050i 0.1634 + 0.3246i 0.1707 + 0.3976i 0.6290 + 0.1393i 0.3734 + 0.2560i w39 −0.1707 + 0.3028i 0.1636 + 0.3208i 0.1700 + 0.3962i 0.6494 + 0.1504i 0.3799 + 0.2517i w40 −0.2066 + 0.6354i 0.1779 + 0.6841i 0.1825 + 0.7468i 1.3127 + 0.1240i 0.7502 + 0.1138i w41 −0.2071 + 0.6404i 0.1828 + 0.6845i 0.1898 + 0.7378i 0.9572 + 0.4344i 0.7325 + 0.2088i w42 −0.2071 + 0.6404i 0.1745 + 0.6828i 0.1588 + 0.7521i 1.2403 + 0.2631i 0.6473 + 0.1138i w43 −0.2075 + 0.6460i 0.1793 + 0.6S29i 0.1666 + 0.7410i 1.0254 + 0.4130i 0.6339 + 0.1702i w44 −0.3219 + 0.5854i 0.3547 + 0.6009i 0.3967 + 0.6104i 0.6096 + 0.4214i 0.5902 + 0.4857i w45 −0.3250 + 0.5893i 0.3593 + 0.6011i 0.3971 + 0.6093i 0.6773 + 0.4284i 0.6355 + 0.4185i w46 −0.3250 + 0.5893i 0.3576 + 0.5990i 0.4022 + 0.5964i 0.5995 + 0.4102i 0.4968 + 0.3947i w47 −0.3285 + 0.5937i 0.3624 + 0.5994i 0.4020 + 0.5952i 0.6531 + 0.4101i 0.5231 + 0.3644i w48 −0.3085 + 0.1570i 0.2697 + 0.1443i 0.2177 + 0.1304i 0.1250 + 0.1153i 0.0872 + 0.1390i w49 −0.3061 + 0.1564i 0.2704 + 0.1433i 0.2168 + 0.1304i 0.1252 + 0.1158i 0.0871 + 0.1392i w50 −0.3061 + 0.1564i 0.2644 + 0.1442i 0.2184 + 0.1305i 0.1245 + 0.1152i 0.0928 + 0.3970i w51 −0.3037 + 0.1557i 0.2650 + 0.1432i 0.2174 + 0.1306i 0.1247 + 0.1156i 0.0937 + 0.3973i w52 −0.3073 + 0.1726i 0.2763 + 0.1638i 0.2264 + 0.1539i 0.3768 + 0.1244i 0.1053 + 0.1494i w53 −0.3050 + 0.1716i 0.2768 + 0.1626i 0.2255 + 0.1538i 0.3707 + 0.1237i 0.1052 + 0.1495i w54 −0.3050 + 0.1716i 0.2715 + 0.1630i 0.2269 + 0.1542i 0.3779 + 0.1260i 0.1938 + 0.3621i w55 −0.3028 + 0.1707i 0.2719 + 0.1618i 0.2260 + 0.1541i 0.3717 + 0.1252i 0.1909 + 0.3627i w56 −0.6354 + 0.2066i 0.6488 + 0.1696i 0.6100 + 0.1390i 0.1161 + 0.3693i 0.1091 + 0.7656i w57 −0.6404 + 0.2071i 0.6462 + 0.1706i 0.6146 + 0.1379i 0.1157 + 0.3645i 0.1699 + 0.7537i w58 −0.6404 + 0.2071i 0.6456 + 0.1745i 0.5973 + 0.1418i 0.1176 + 0.3469i 0.1054 + 0.5979i w59 −0.6460 + 0.2075i 0.6431 + 0.1753i 0.6021 + 0.1407i 0.1171 + 0.3424i 0.1230 + 0.5949i w60 −0.5854 + 0.3219i 0.5854 + 0.3186i 0.5671 + 0.2989i 0.3530 + 0.3899i 0.4294 + 0.6363i w61 −0.5893 + 0.3250i 0.5862 + 0.3167i 0.5699 + 0.2952i 0.3422 + 0.3808i 0.3744 + 0.6744i w62 −0.5893 + 0.3250i 0.5864 + 0.3275i 0.5622 + 0.3050i 0.3614 + 0.3755i 0.3224 + 0.5236i w63 −0.5937 + 0.3285i 0.5873 + 0.3254i 0.5648 + 0.3012i 0.3509 + 0.3656i 0.3016 + 0.5347i w/code rate 10/15 11/15 12/15 13/15 w0 1.6097 + 0.1548i 0.1512 + 1.5761i 1.1980 + 1.1541i 1.1014 + 1.1670i w1 1.5549 + 0.4605i 0.4510 + 1.5251i 0.9192 + 1.2082i 0.8557 + 1.2421i w2 1.3226 + 0.1290i 0.1269 + 1.3158i 1.2778 + 0.8523i 1.2957 + 0.8039i w3 1.2772 + 0.3829i 0.3773 + 1.2728i 1.0390 + 0.9253i 1.0881 + 0.8956i w4 1.2753 + 1.0242i 1.0045 + 1.2537i 0.6057 + 1.2200i 0.5795 + 1.2110i w5 1.4434 + 0.7540i 0.7386 + 1.4164i 0.7371 + 1.4217i 0.6637 + 1.4215i w6 1.0491 + 0.8476i 0.8392 + 1.0508i 0.6678 + 1.0021i 0.6930 + 1.0082i w7 1.1861 + 0.6253i 0.6172 + 1.1844i 0.8412 + 0.9448i 0.8849 + 0.9647i w8 0.9326 + 0.0970i 0.0939 + 0.9317i 1.2128 + 0.5373i 1.2063 + 0.5115i w9 0.3962 + 0.2804i 0.2781 + 0.8975i 1.0048 + 0.5165i 1.0059 + 0.4952i w10 1.1044 + 0.1102i 0.1093 + 1.1073i 1.4321 + 0.6343i 1.4171 + 0.5901i w11 1.0648 + 0.3267i 0.3230 + 1.0701i 1.0245 + 0.7152i 1.0466 + 0.6935i w12 0.7325 + 0.6071i 0.6074 + 0.7404i 0.6384 + 0.6073i 0.6639 + 0.6286i w13 0.8260 + 0.4559i 0.4499 + 0.8324i 0.8175 + 0.5684i 0.8353 + 0.5851i w14 0.8744 + 0.7153i 0.7128 + 0.8852i 0.6568 + 0.7801i 0.6879 + 0.8022i w15 0.9882 + 0.5300i 0.5259 + 0.9957i 0.8311 + 0.7459i 0.8634 + 0.7622i w16 0.1646 + 1.6407i 1.6163 + 0.1632i 0.1349 + 1.4742i 0.1213 + 1.4366i w17 0.4867 + 1.5743i 1.5486 + 0.4813i 0.1105 + 1.2309i 0.1077 + 1.2098i w13 0.1363 + 1.3579i 1.3596 + 0.1381i 0.0634 + 0.9796i 0.0651 + 0.9801i w19 0.4023 + 1.3026i 1.3018 + 0.4076i 0.1891 + 1.0198i 0.2009 + 1.0115i w20 1.0542 + 1.2584i 1.2347 + 1.0376i 0.4142 + 1.4461i 0.3764 + 1.4264i w21 0.7875 + 1.4450i 1.4193 + 0.7769i 0.3323 + 1.2279i 0.3237 + 1.2130i w22 0.3637 + 1.0407i 1.0335 + 0.8734i 0.4998 + 0.9827i 0.5205 + 0.9814i w23 0.6502 + 1.1951i 1.1905 + 0.6562i 0.3467 + 1.0202i 0.3615 + 1.0163i w24 0.0982 + 0.9745i 0.9948 + 0.1006i 0.0680 + 0.6501i 0.0715 + 0.6596i w25 0.2842 + 0.9344i 0.9492 + 0.2955i 0.2016 + 0.6464i 0.2116 + 0.6597i w26 0.1142 + 1.1448i 1.1596 + 0.1180i 0.0719 + 0.8075i 0.0729 + 0.8131i w27 0.3335 + 1.0973i 1.1091 + 0.3468i 0.2088 + 0.8146i 0.2158 + 0.8246i w28 0.6062 + 0.7465i 0.7476 + 0.6205i 0.4809 + 0.6296i 0.5036 + 0.6467i w29 0.4607 + 0.8538i 0.8645 + 0.4711i 0.3374 + 0.6412i 0.3526 + 0.6572i w30 0.7263 + 0.8764i 0.8777 + 0.7384i 0.4955 + 0.8008i 0.5185 + 0.S086i w31 0.5450 + 1.0067i 1.0128 + 0.5570i 0.3431 + 0.8141i 0.3593 + 0.8245i w32 0.2655 + 0.0746i 0.0723 + 0.2545i 1.2731 + 0.1108i 1.2545 + 0.1010i w33 0.2664 + 0.0759i 0.0761 + 0.2566i 1.0794 + 0.0977i 1.0676 + 0.0956i w34 0.4571 + 0.0852i 0.0729 + 0.4512i 1.5126 + 0.1256i 1.4782 + 0.1167i w35 0.4516 + 0.1062i 0.1145 + 0.4434i 0.9029 + 0.0853i 0.8981 + 0.0882i w36 0.2559 + 0.1790i 0.2114 + 0.2443i 0.5429 + 0.0694i 0.5518 + 0.0690i w37 0.2586 + 0.1772i 0.2043 + 0.2508i 0.6795 + 0.0559i 0.6903 + 0.0552i w38 0.3592 + 0.2811i 0.3079 + 0.3516i 0.5628 + 0.1945i 0.5742 + 0.1987i w39 0.3728 + 0.2654i 0.2673 + 0.3812i 0.7326 + 0.1410i 0.7374 + 0.1564i w40 0.7706 + 0.0922i 0.0813 + 0.7729i 1.2283 + 0.3217i 1.2378 + 0.3049i w41 0.7407 + 0.2260i 0.2345 + 0.7412i 1.0269 + 0.3261i 1.0518 + 0.3032i w42 0.6180 + 0.0927i 0.0756 + 0.6187i 1.4663 + 0.3716i 1.4584 + 0.3511i w43 0.6019 + 0.1658i 0.1815 + 0.5946i 0.9085 + 0.2470i 0.9107 + 0.2603i w44 0.6007 + 0.4980i 0.5118 + 0.6039i 0.6160 + 0.4549i 0.6321 + 0.4729i w45 0.6673 + 0.3928i 0.3821 + 0.6808i 0.7818 + 0.4247i 0.7880 + 0.4392i w46 0.4786 + 0.3935i 0.4145 + 0.4734i 0.5938 + 0.3170i 0.6045 + 0.3274i w47 0.5176 + 0.3391i 0.3230 + 0.5340i 0.7600 + 0.2850i 0.7629 + 0.2965i w48 0.0757 + 0.1003i 0.0858 + 0.0734i 0.0595 + 0.0707i 0.0596 + 0.0739i w49 0.0753 + 0.1004i 0.0860 + 0.0722i 0.1722 + 0.0706i 0.1767 + 0.0731i w50 0.0777 + 0.4788i 0.5212 + 0.0693i 0.0599 + 0.2119i 0.0612 + 0.2198i w51 0.0867 + 0.4754i 0.5118 + 0.0896i 0.1748 + 0.2114i 0.1815 + 0.2192i w52 0.1023 + 0.2243i 0.2611 + 0.0940i 0.4134 + 0.0701i 0.4218 + 0.0715i w53 0.1010 + 0.2242i 0.2628 + 0.0898i 0.2935 + 0.0705i 0.2978 + 0.0725i w54 0.1950 + 0.3919i 0.4159 + 0.2145i 0.4231 + 0.2066i 0.4337 + 0.2115i w55 0.1881 + 0.3969i 0.4287 + 0.1928i 0.2979 + 0.2100i 0.3057 + 0.2167i w56 0.0930 + 0.8122i 0.8427 + 0.0848i 0.0638 + 0.5002i 0.0667 + 0.5124i w57 0.2215 + 0.7840i 0.8036 + 0.2443i 0.1905 + 0.4966i 0.2008 + 0.5095i w58 0.0937 + 0.6514i 0.6933 + 0.0788i 0.0612 + 0.3552i 0.0625 + 0.3658i w59 0.1540 + 0.6366i 0.6649 + 0.1793i 0.1810 + 0.3533i 0.1899 + 0.3642i w60 0.4810 + 0.6306i 0.6321 + 0.5048i 0.4630 + 0.4764i 0.4818 + 0.4946i w61 0.3856 + 0.7037i 0.7279 + 0.3892i 0.3231 + 0.4895i 0.3380 + 0.5050i w62 0.3527 + 0.5230i 0.5265 + 0.3752i 0.4416 + 0.3397i 0.4571 + 0.3499i w63 0.3100 + 0.5559i 0.5885 + 0.3076i 0.3083 + 0.3490i 0.3216 + 0.3599i C4) 1024-QQAM NUC

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0.5425 + 0.5800i w179 0.5543 + 0.4350i 0.4587 + 0.5943i 0.6118 + 0.4591i 0.4782 + 0.6270i w180 0.7304 + 0.4323i 0.3616 + 0.7417i 0.7598 + 0.3644i 0.3667 + 0.7681i w181 0.6740 + 0.4376i 0.4272 + 0.7061i 0.7208 + 0.4365i 0.4442 + 0.7324i w182 0.7290 + 0.4235i 0.3335 + 0.6719i 0.6887 + 0.3313i 0.3373 + 0.6966i w183 0.6729 + 0.4278i 0.3844 + 0.6452i 0.6540 + 0.3956i 0.4091 + 0.6660i w184 0.5295 + 0.3071i 0.3879 + 0.4475i 0.4535 + 0.4040i 0.4524 + 0.4675i w185 0.5477 + 0.3068i 0.3776 + 0.4571i 0.4792 + 0.3721i 0.4008 + 0.5029i w186 0.5297 + 0.3097i 0.4424 + 0.5022i 0.5082 + 0.4618i 0.4977 + 0.5232i w187 0.5479 + 0.3092i 0.4191 + 0.5225i 0.5468 + 0.4147i 0.4386 + 0.5659i w188 0.7223 + 0.3006i 0.2791 + 0.5203i 0.5417 + 0.2697i 0.2843 + 0.5533i w189 0.6675 + 0.3031i 0.2927 + 0.5124i 0.5222 + 0.3074i 0.3447 + 0.5290i w190 0.7215 + 0.3022i 0.3056 + 0.5933i 0.6164 + 0.2990i 0.3097 + 0.6251i w191 0.6671 + 0.3051i 0.3356 + 0.5770i 0.5877 + 0.3530i 0.3756 + 0.5986i w192 0.0563 + 0.0613i 0.1787 + 0.0387i 0.0499 + 0.2055i 0.1765 + 0.0357i w193 0.0563 + 0.0613i 0.1788 + 0.0380i 0.0509 + 0.2049i 0.2273 + 0.0412i w194 0.0563 + 0.0613i 0.2844 + 0.0453i 0.0399 + 0.2999i 0.3691 + 0.0298i w195 0.0563 + 0.0614i 0.2853 + 0.0454i 0.0401 + 0.3002i 0.2911 + 0.0346i w196 0.1675 + 0.0612i 0.0534 + 0.0447i 0.0338 + 0.1277i 0.1117 + 0.0289i w197 0.1675 + 0.0612i 0.0536 + 0.0460i 0.0333 + 0.1274i 0.0382 + 0.0258i w198 0.1675 + 0.0612i 0.3158 + 0.0720i 0.0946 + 0.3296i 0.3643 + 0.0969i w199 0.1675 + 0.0613i 0.3142 + 0.0743i 0.0941 + 0.3290i 0.3097 + 0.0874i w200 0.0566 + 0.1845i 0.5089 + 0.0486i 0.0419 + 0.5217i 0.5699 + 0.0285i w201 0.0565 + 0.1845i 0.5084 + 0.0527i 0.0573 + 0.5200i 0.5487 + 0.0713i w202 0.0565 + 0.1839i 0.4222 + 0.0426i 0.0352 + 0.4266i 0.4393 + 0.0368i w203 0.0565 + 0.1839i 0.4214 + 0.0422i 0.0371 + 0.4263i 0.4320 + 0.0488i w204 0.1681 + 0.1844i 0.4864 + 0.1436i 0.1548 + 0.4959i 0.5432 + 0.1831i w205 0.1682 + 0.1844i 0.4880 + 0.1415i 0.1395 + 0.5007i 0.5356 + 0.1368i w206 0.1681 + 0.1838i 0.4019 + 0.1047i 0.1182 + 0.4140i 0.4214 + 0.1199i w207 0.1682 + 0.1838i 0.4023 + 0.1060i 0.1141 + 0.4149i 0.4661 + 0.1312i w208 0.4059 + 0.0609i 0.1614 + 0.0913i 0.1305 + 0.1866i 0.1610 + 0.1081i w209 0.4022 + 0.0608i 0.1612 + 0.0906i 0.1305 + 0.1856i 0.2042 + 0.1262i w210 0.4058 + 0.0609i 0.2488 + 0.1623i 0.1936 + 0.2536i 0.2973 + 0.2229i w211 0.4022 + 0.0609i 0.2488 + 0.1628i 0.1935 + 0.2542i 0.2455 + 0.1743i w212 0.2835 + 0.0610i 0.0659 + 0.0671i 0.0466 + 0.0504i 0.1014 + 0.0859i w213 0.2840 + 0.0610i 0.0654 + 0.0671i 0.0470 + 0.0503i 0.0352 + 0.0762i w214 0.2835 + 0.0611i 0.2866 + 0.1555i 0.1636 + 0.3085i 0.3374 + 0.1692i w215 0.2840 + 0.0610i 0.2872 + 0.1564i 0.1629 + 0.3022i 0.2914 + 0.1413i w216 0.4066 + 0.1836i 0.3982 + 0.3161i 0.3355 + 0.4003i 0.4468 + 0.3693i w217 0.4027 + 0.1835i 0.4005 + 0.3136i 0.3227 + 0.4102i 0.4545 + 0.3221i w218 0.4066 + 0.1831i 0.3333 + 0.2586i 0.2750 + 0.3318i 0.3539 + 0.2625i w219 0.4027 + 0.1830i 0.3330 + 0.2579i 0.2722 + 0.3342i 0.3954 + 0.2905i w220 0.2842 + 0.1841i 0.4515 + 0.2301i 0.2346 + 0.4627i 0.5230 + 0.2380i w221 0.2847 + 0.1841i 0.4493 + 0.2334i 0.2495 + 0.4548i 0.4892 + 0.2652i w222 0.2842 + 0.1835i 0.3681 + 0.1950i 0.2020 + 0.3827i 0.3945 + 0.1942i w223 0.2847 + 0.1836i 0.3693 + 0.1954i 0.2054 + 0.3804i 0.4344 + 0.2178i w224 0.0567 + 0.4509i 0.8318 + 0.0442i 0.0421 + 0.5420i 0.8751 + 0.0419i w225 0.0566 + 0.4510i 0.8239 + 0.1192i 0.1247 + 0.8342i 0.8676 + 0.1257i w226 0.0568 + 0.4376i 0.7562 + 0.0454i 0.0384 + 0.7648i 0.7986 + 0.0381i w227 0.0568 + 0.4376i 0.7500 + 0.1031i 0.1132 + 0.7575i 0.7927 + 0.1144i w223 0.1691 + 0.4514i 0.7842 + 0.2789i 0.2858 + 0.7945i 0.8293 + 0.2896i w229 0.1691 + 0.4515i 0.8063 + 0.2068i 0.2067 + 0.8184i 0.3523 + 0.2091i w230 0.1694 + 0.4387i 0.7167 + 0.2465i 0.2585 + 0.7216i 0.7578 + 0.2626i w231 0.1693 + 0.4387i 0.7319 + 0.1921i 0.1886 + 0.7429i 0.7786 + 0.1893i w232 0.0568 + 0.3093i 0.5933 + 0.0507i 0.0401 + 0.6073i 0.6439 + 0.0332i w233 0.0568 + 0.3093i 0.5924 + 0.0642i 0.0800 + 0.6023i 0.6363 + 0.0929i w234 0.0567 + 0.3125i 0.6712 + 0.0500i 0.0378 + 0.6868i 0.7212 + 0.0348i w235 0.0568 + 0.3125i 0.6689 + 0.0819i 0.0991 + 0.6806i 0.7163 + 0.1032i w236 0.1690 + 0.3097i 0.5683 + 0.1760i 0.1934 + 0.5729i 0.6122 + 0.2069i w237 0.1690 + 0.3097i 0.5720 + 0.1640i 0.1567 + 0.5853i 0.6268 + 0.1493i w238 0.1690 + 0.3128i 0.6410 + 0.2092i 0.2285 + 0.6450i 0.6353 + 0.2349i w239 0.1689 + 0.3128i 0.6491 + 0.1788i 0.1719 + 0.6651i 0.7046 + 0.1699i w240 0.4128 + 0.4506i 0.6189 + 0.5514i 0.5654 + 0.6265i 0.6521 + 0.5790i w241 0.4075 + 0.4506i 0.6665 + 0.4960i 0.5027 + 0.6784i 0.7094 + 0.5138i w242 0.4124 + 0.4395i 0.5667 + 0.4955i 0.5121 + 0.5697i 0.6013 + 0.5265i w243 0.4073 + 0.4395i 0.6028 + 0.4546i 0.4555 + 0.6162i 0.6524 + 0.4667i w244 0.2866 + 0.4518i 0.7506 + 0.3592i 0.3624 + 0.7630i 0.7984 + 0.3673i w245 0.2875 + 0.4520i 0.7156 + 0.4239i 0.4347 + 0.7245i 0.7580 + 0.4428i w246 0.2868 + 0.4399i 0.6809 + 0.3299i 0.3286 + 0.6923i 0.7298 + 0.3325i w247 0.2876 + 0.4400i 0.6544 + 0.3786i 0.3924 + 0.6581i 0.6964 + 0.4016i w248 0.4087 + 0.3094i 0.4599 + 0.3752i 0.3972 + 0.4584i 0.5005 + 0.4181i w249 0.4044 + 0.3093i 0.4675 + 0.3645i 0.3657 + 0.4832i 0.5359 + 0.3697i w250 0.4087 + 0.3122i 0.5121 + 0.4317i 0.4563 + 0.5135i 0.5509 + 0.4732i w251 0.4044 + 0.3121i 0.5325 + 0.4077i 0.4093 + 0.5514i 0.5953 + 0.4196i w252 0.2855 + 0.3099i 0.5290 + 0.2684i 0.2657 + 0.5428i 0.5896 + 0.2686i w253 0.2861 + 0.3100i 0.5240 + 0.2788i 0.2996 + 0.5248i 0.5635 + 0.3210i w254 0.2855 + 0.3129i 0.6016 + 0.2998i 0.2962 + 0.6193i 0.6601 + 0.3001i w255 0.2861 + 0.3130i 0.5877 + 0.3267i 0.3481 + 0.5915i 0.6312 + 0.3619i

Next, a definition of the QQAM constellation shall be provided. Each input cell word (y₀, . . . , y_(m-1)) shall be modulated using a non-uniform QQAM constellations to give a constellation point zq prior to normalization, where m corresponds to the number of bits per QAM symbol m=log₂(M). The vector of complex constellation points x_(0 . . . M-1) for all combinations of the input bits y_(0 . . . m-1) (corresponding to the decimal values 0 to M−1) are given in the above shown tables for the various constellation sizes depending on the QQAM position vector w_(0 . . . b-1), which defines the constellation point positions of a first quarter of the non-uniform constellation. The length b of the QQAM position vector w is defined by b=M/4. The QQAM position vector defines a first quarter of the constellation, namely the constellation points with the decimal values 0 (y_(0 . . . m)=0000 for the example of a 16-QQAM) to b−1 (y_(0 . . . m)=0011 for the example of a 16-QQAM), while the remaining constellation points are derived as follows: x _(0 . . . b-1) =w _(0 . . . b-1)  (first quarter) x _(b . . . 2b-1)=conj(w _(0 . . . b-1))  (second quarter) x _(2b . . . 3b-1)=−conj(w _(0 . . . b-1))  (third quarter) x _(3b . . . 4b-1) =−w _(0 . . . b-1)  (fourth quarter) with conj being the complex conjugate. For example, the corresponding constellation point z_(q) for a 16-QQAM defined by the QQAM position vector (we 3)=(0.2663+0.4530i 0.4530+0.2663i 0.5115+1.2092i 1.2092+0.5115i) and the input cell word (y₀ . . . y_(m-1))=(1100) is x₁₂=−w₀=−0.2663 −0.4530i. The complete constellation for this NUC position vector (optimized for code rate 6/15) is shown in the FIG. 7 with all input cell words marked at the corresponding constellation points.

The QQAM and the N²-NUC constellations have been defined in such a way that the bit-wise mutual information is increasing with the bit position, i.e. the MSB (leftmost bit label) carries the largest mutual information, while the LSB (rightmost bit label) carries the least mutual information. As mentioned above the constellation position vector w as defined herein does not necessarily contain the constellation points of a quadrant, e.g. the first quadrant, of the constellation, but could also contain the constellation points of any of the four quadrants or a quarter of constellation points which are not all located in a single quadrant. Due to the symmetry this leads to constellations with a different bit mapping but with identical performance. The constellation position vector w in the tables defined herein should therefore be considered as an example for all four symmetric constellations with different bit mapping but identical performance.

Using N²-QAM constellations it is meaningful from an information theoretic point of view to use high constellation orders, since these constellations offer more degrees of freedom for the optimization and perform closer to the Shannon capacity as depicted in FIG. 8 . However, with increasing constellation size the complexity for demapping in the receiver also increases. Since for large N²QAM constellations many constellation points are very close to each other in the complex plane it is proposed in Jonathan Stott, “CM and BICM limits for rectangular constellations”, DVB document server, document TM-MIMO0007. August 2012, to “condense” non-uniform constellations by means of forcing particular constellation points to have the same position before the optimization process, accepting a small performance loss compared to its “mother constellation”. Such constellations are called there “ConQAM” (condensed QAM) and the condensed constellations of QQAM constellations are called there “ConQQAM”. This provides a reduced complexity during the optimization process, since fewer degrees of freedom have to be optimized and a reduced complexity for demapping in the receiver, due to the reduced number of “effective” constellation points. In the above mentioned document of Jonathan Stott a condensed 16 kQAM has been presented with only 3600 remaining constellation point positions, offering a good performance in the SNR region from 20 to 25 dB.

When the condensation is performed before the optimization, assumptions must be made, how a good performing constellation may look like (i.e. which particular points are condensed and which not). This requires a deep analysis for high constellation sizes. Based on these assumptions of the chosen structure of the constellation, the optimization is carried out over an SNR region with the corresponding number of constellation points (e.g. 268 condensed constellation points instead of 1024). The drawback of this approach is that the optimal structure of the constellation practically changes for each SNR value, which cannot be taken into account. That is the resulting ConQAM constellation with a fixed number of constellations points is not optimal over a broad SNR range. Therefore different structures are herein derived and optimized.

An improved alternative to condensing the constellation before the optimization is the reduction of the constellation points after the optimization which is proposed according to the present disclosure. The optimization of all degrees of freedom of the N²-QAM constellation is thus required, but several advantages are obtained. When performing the condensation after the optimization, a constellation requiring the minimum required number of constellation points can be derived to offer a desired performance. This allows for a seamless change of the required number of constellation points over the SNR range, which leads to a reduction of the number of constellation points compared to the approach proposed in the above mentioned document of Jonathan Stott. This approach will be called dynamic condensation, since it is carried out for each SNR point individually. This approach is outlined for the N²-QAM case in the following.

An example of the algorithm is shown in FIG. 9 for 17 constellation points of a PAM constellation: The constellation points with a distance smaller than the threshold t result in a group of constellation points, i.e. are condensed to a single constellation point position. In the end only 6 constellation points are remaining. Of course, the algorithm can analogously be extended to the 2D-case as will be briefly explained below.

The required number of constellation points of the dynamic approach is clearly lower, in addition guaranteeing a maximum performance penalty with respect to the mother constellation. This leads to a reduced number of constellation points, further reducing the complexity in the demapper.

In an embodiment modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from a group B of constellations comprising one or more of the following constellations defined by the constellation position vector u of length v=sqrt(M)/2−1, wherein in one or more constellation position vectors of the constellations from group B two or more constellation positions are identical resulting from a condensation of preliminary constellation positions optimized before.

The following non-uniform constellations comprised in group B are proposed:

B) condensed M-QAM non-uniform constellations of group B:

B1) 256-ConQAM NUC

code rate u 6/15 7/15 8/15 9/15 u1 1 1 1 1 u2 2.2838 2.6712 2.6867 2.6537 u3 2.2838 2.6712 3.0876 3.3114 u4 4.2938 4.6718 4.8578 4.9478 u5 4.7065 5.3606 5.9639 6.2057 u6 6.5754 7.387 7.9599 8.1648 u7 8.6463 9.7667 10.4826 10.6773 B2) 1024-ConQAM NUC

u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 u1 1 1 1 1 1 1 u2 1 1 1 2.0888 2.5646 2.7751 u3 1 1 1 2.0888 2.5646 2.7751 u4 2.4714 2.7615 2.6688 3.9929 4.55 4.8138 u5 2.4714 2.7615 2.6688 3.9929 4.55 4.8138 u6 2.4714 2.7615 3.2331 5.3856 6.3402 6.878 u7 2.4714 2.7615 3.2331 5.3856 6.3402 6.878 u8 4.4808 4.735 4.9096 7.5592 8.6388 8.8948 u9 4.4808 4.735 4.9096 7.5592 8.6388 9.9366 u10 4.9849 5.5675 6.1134 9.3349 10.5992 11.5652 u11 4.9849 5.5675 6.1134 9.8406 11.6813 13.0327 u12 6.9079 7.3286 7.7771 11.9228 13.7177 15.0299 u13 6.9079 7.819 8.5339 13.3931 15.6096 17.0928 u14 8.576 9.5608 10.3128 15.8944 18.2724 19.7629 u15 10.4874 11.6967 12.5661 19.1546 21.7521 23.229 B3) 4096-ConQAM NUC

u/code rate 5/15 6/15 7/15 8/15 9/15 10/15 11/15 12/15 u1 1 1 1 1 1 1 1 1 u2 1 1 1 1 1 2.2716 2.7361 2.8773 u3 1 1 1 1 1 2.2716 2.7361 2.8773 u4 1 1 2.039 2.5335 2.7564 4.1311 4.6854 4.8671 u5 1 1 2.039 2.5335 2.7564 4.1311 4.6854 4.8671 u6 1 1 2.039 2.5335 2.7564 5.4628 6.4578 6.804 u7 1 1 2.039 2.5335 2.7564 5.4628 6.4578 6.804 u8 2.632 2.6741 3.9335 4.5125 4.7808 7.4745 8.5074 8.5047 u9 2.632 2.6741 3.9335 4.5125 4.7808 7.4745 8.5074 9.3098 u10 2.632 2.6741 3.9335 4.5125 4.7808 8.9342 10.3862 10.538 u11 2.632 2.6741 3.9335 4.5125 4.7808 8.9342 10.3862 11.4018 u12 2.632 3.2045 5.2825 6.2741 6.7922 10.9126 12.5386 12.7315 u13 2.632 3.2045 5.2825 6.2741 6.7922 10.9126 12.5386 13.671 u14 2.632 3.2045 5.2825 6.2741 6.7922 12.5824 14.7019 14.9604 u15 2.632 3.2045 5.2825 6.2741 6.7922 12.5824 14.7019 15.9972 u16 4.6312 4.8953 7.4395 8.5533 8.7969 14.7996 17.2181 17.4143 u17 4.6312 4.8953 7.4395 8.5533 8.7969 14.7996 17.2181 18.5456 u18 4.6312 4.8953 7.4395 8.5533 9.7606 16.7338 19.7895 19.9493 u19 4.6312 4.8953 7.4395 8.5533 9.7606 16.7338 19.7895 21.1969 u20 5.2613 6.077 9.1955 10.4959 11.3919 19.1103 22.0532 22.741 u21 5.2613 6.077 9.1955 10.4959 11.3919 19.1103 23.3808 24.129 u22 5.2613 6.077 9.6429 11.4947 12.8073 21.6213 25.1169 25.7517 u23 5.2613 6.077 9.6429 11.4947 12.8073 21.6213 26.6987 27.3256 u24 7.0507 7.7374 11.7079 13.5198 14.7968 24.1315 28.6925 29.1651 u25 7.0507 7.7374 11.7079 13.5198 14.7968 25.5633 30.5564 30.9655 u26 7.4269 8.4608 13.0989 15.1128 16.5067 27.6414 32.7592 33.0129 u27 7.4269 8.4608 13.0989 15.6457 17.5025 29.6431 35.04 35.1483 u28 8.869 9.9898 15.1744 17.6029 19.3416 32.2532 37.7468 37.6166 u29 9.1641 10.497 16.2183 19.0971 21.1053 35.0366 40.6819 40.3063 u30 10.5034 12.1004 18.4967 21.5328 23.5673 38.5846 44.2782 43.5432 u31 12.3418 14.1335 21.4623 24.7544 26.8076 43.1809 48.8556 47.6401

When optimizing non-uniform QAM constellations, some of the constellation points tend to merge. This can be exploited by deliberately merging constellation points that lie close to each other in order to reduce the complexity in the QAM demapper (but also in the QAM mapper), by simplifying the calculation of the soft-decision log likelihood ratios (LLRs). Such constellations are called condensed QAM constellations. If chosen carefully, the loss in performance compared to non-condensed non-uniform constellations can be neglected. As an example, the 1024-QQAM constellation optimized for code rate 6/15 can be condensed to 268 constellation point positions reducing the demapping complexity while retaining the performance.

In an embodiment the modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from a group D of constellations comprising one or more of the following constellations, wherein the constellation points are defined by a constellation position vector w_(0 . . . b-1) with b=M/4, wherein

a first quarter of the constellation points x_(0 . . . b-1) are defined as x_(0 . . . b-1)=w_(0 . . . b-1),

a second quarter of the constellation points x_(b . . . 2b-1) are defined as x_(b . . . 2b-1)=conj(w_(0 . . . b-1)),

a third quarter of the constellation points x_(2b . . . 3b-1) are defined as x_(2b . . . 3b-1)=−conj(w_(0 . . . b-1)), and a fourth quarter of the constellation points x_(3b . . . 4b-1) are defined as x_(3b . . . 4b-1)=−w_(0 . . . b1), wherein conj is the complex conjugate,

wherein in one or more constellation position vectors of the constellations from group D two or more constellation positions are identical resulting from a condensation of preliminary constellation positions optimized before.

The following non-uniform constellations comprised in group D are proposed:

D) condensed M-QAM non-uniform constellations of group D:

D1) 64-ConQQAM NUC

code rate w 5/15 6/15 7/15 8/15 9/15 w0 1.0257 + 0.5960i 0.5656 + 0.9499i 0.2925 + 1.4892i 0.2920 + 1.4827i 0.2978 + 1.4669i w1 1.2181 + 0.7476i 0.2750 + 1.0676i 0.8449 + 1.2622i 0.8411 + 1.2563i 0.8421 + 1.2355i w2 1.1509 + 0.3069i 0.8202 + 1.2651i 0.2351 + 1.0196i 0.2174 + 1.0211i 0.2135 + 1.0389i w3 1.3888 + 0.3325i 0.3011 + 1.4529i 0.5555 + 0.8926i 0.5702 + 0.8798i 0.6055 + 0.8654i w4 0.5961 + 1.0257i 0.9500 + 0.5641i 1.4892 + 0.2925i 1.4827 + 0.2920i 1.4685 + 0.2859i w5 0.7476 + 1.2181i 1.0666 + 0.2744i 1.2622 + 0.8449i 1.2563 + 0.8410i 1.2516 + 0.8201i w6 0.3069 + 1.1510i 1.2657 + 0.8178i 1.0196 + 0.2351i 1.0211 + 0.2174i 1.0279 + 0.1981i w7 0.3325 + 1.3888i 1.4521 + 0.3005i 0.8926 + 0.5555i 0.8798 + 0.5702i 0.8857 + 0.5642i w8 0.4969 + 0.2685i 0.2717 + 0.5512i 0.1635 + 0.3025i 0.1583 + 0.3034i 0.1545 + 0.3665i w9 0.4123 + 0.2376i 0.2717 + 0.5512i 0.1635 + 0.3025i 0.1583 + 0.3034i 0.1545 + 0.3665i w10 0.4969 + 0.2685i 0.2020 + 0.3572i 0.2075 + 0.6586i 0.1871 + 0.6855i 0.1756 + 0.7261i w11 0.4123 + 0.2376i 0.2020 + 0.3572i 0.3354 + 0.6030i 0.3563 + 0.6126i 0.4023 + 0.6126i w12 0.2685 + 0.4969i 0.5503 + 0.2716i 0.3025 + 0.1635i 0.3034 + 0.1583i 0.2731 + 0.1455i w13 0.2376 + 0.4123i 0.5503 + 0.2716i 0.3025 + 0.1635i 0.3034 + 0.1583i 0.2731 + 0.1455i w14 0.2685 + 0.4969i 0.3561 + 0.2021i 0.6586 + 0.2075i 0.6855 + 0.1871i 0.6840 + 0.1578i w15 0.2376 + 0.4123i 0.3561 + 0.2021i 0.6030 + 0.3354i 0.6126 + 0.3563i 0.6145 + 0.3556i D2) 256-ConQQAM NUC

w/code 5/15 6/15 7/15 8/15 9/15 10/15 11/15 w0 −0.3495 + 1.7549i 0.6800 + 1.6926i 0.7280 + 1.6384i 1.0804 + 1.3788i 1.6350 + 0.1593i 1.6097 + 0.1548i 0.1512 + 1.5761i w1 −0.2804 + 1.4293i 0.3911 + 1.3645i 0.4787 + 1.3492i 1.0487 + 0.9862i 1.5776 + 0.4735i 1.5549 + 0.4605i 0.4510 + 1.5251i w2 −0.2804 + 1.4293i 0.2191 + 1.7524i 0.2417 + 1.7872i 1.6464 + 0.7428i 1.3225 + 0.1320i 1.3226 + 0.1290i 0.1269 + 1.3158i w3 −0.2635 + 1.3614i 0.2274 + 1.4208i 0.1966 + 1.4478i 1.3245 + 0.9414i 1.2742 + 0.3922i 1.2772 + 0.3829i 0.3773 + 1.2728i w4 −0.9918 + 1.4880i 0.8678 + 1.2487i 0.9185 + 1.2490i 0.7198 + 1.2427i 1.2901 + 1.0495i 1.2753 + 1.0242i 1.0045 + 1.2537i w5 −0.8109 + 1.21161 0.7275 + 1.1667i 0.7448 + 1.1524i 0.8106 + 1.0040i 1.4625 + 0.7740i 1.4434 + 0.7540i 0.7386 + 1.4164i w6 −0.8109 + 1.2116i 0.8747 + 1.0470i 0.9536 + 0.9516i 0.5595 + 1.0317i 1.0382 + 0.8623i 1.0491 + 0.8476i 0.8392 + 1.0508i w7 −0.7744 + 1.1516i 0.7930 + 1.0406i 0.8912 + 0.9461i 0.6118 + 0.9722i 1.1794 + 0.6376i 1.1861 + 0.6253i 0.6172 + 1.1844i w8 −0.2349 + 1.0113i 0.1978 + 0.9823i 0.2553 + 0.9993i 1.6768 + 0.2002i 0.9430 + 0.1100i 0.9326 + 0.0970i 0.0939 + 0.9317i w9 −0.2349 + 1.0113i 0.2071 + 1.0557i 0.2988 + 1.0689i 0.9997 + 0.6844i 0.9069 + 0.2829i 0.8962 + 0.2804i 0.2781 + 0.8975i w10 −0.2349 + 1.0113i 0.1978 + 0.9823i 0.1656 + 1.0288i 1.4212 + 0.4769i 1.0854 + 0.1139i 1.1044 + 0.1102i 0.1093 + 1.1073i w11 −0.2349 + 1.0113i 0.2071 + 1.0557i 0.1779 + 1.1140i 1.1479 + 0.6312i 1.0441 + 0.3296i 1.0648 + 0.3267i 0.3230 + 1.0701i w12 −0.5497 + 0.8869i 0.5660 + 0.8461i 0.5795 + 0.8287i 0.6079 + 0.6566i 0.7273 + 0.6160i 0.7325 + 0.6071i 0.6074 + 0.7404i w13 −0.5497 + 0.8869i 0.5660 + 0.8461i 0.5795 + 0.8287i 0.7284 + 0.6957i 0.8177 + 0.4841i 0.8260 + 0.4559i 0.4499 + 0.8324i w14 −0.5497 + 0.8869i 0.6073 + 0.8182i 0.6595 + 0.7742i 0.5724 + 0.7031i 0.8504 + 0.7217i 0.8744 + 0.7153i 0.7128 + 0.8852i w15 −0.5497 + 0.8869i 0.5660 + 0.8461i 0.6595 + 0.7742i 0.6302 + 0.7259i 0.9638 + 0.5407i 0.9882 + 0.5300i 0.5259 + 0.9957i w16 −1.7549 + 0.3495i 1.4070 + 0.1790i 1.4079 + 0.1358i 0.1457 + 1.4010i 0.1658 + 1.6747i 0.1646 + 1.6407i 1.6163 + 0.1632i w17 −1.4293 + 0.2804i 1.7227 + 0.2900i 1.7492 + 0.2856i 0.1866 + 1.7346i 0.4907 + 1.6084i 0.4867 + 1.5743i 1.5486 + 0.4813i w18 −1.4293 + 0.2804i 1.3246 + 0.2562i 1.3108 + 0.2733i 0.1174 + 1.1035i 0.1322 + 1.3631i 0.1363 + 1.3579i 1.3596 + 0.1381i w19 −1.3614 + 0.2635i 1.3636 + 0.3654i 1.3393 + 0.4031i 0.1095 + 1.0132i 0.3929 + 1.3102i 0.4023 + 1.3026i 1.3018 + 0.4076i w20 −1.4880 + 0.9918i 1.3708 + 1.2834i 1.3733 + 1.2596i 0.4357 + 1.3636i 1.0646 + 1.2876i 1.0542 + 1.2584i 1.2347 + 1.0376i w21 −1.2116 + 0.8109i 1.6701 + 0.8403i 1.6601 + 0.8198i 0.5853 + 1.6820i 0.7949 + 1.4772i 0.7875 + 1.4450i 1.4193 + 0.7769i w22 −1.2116 + 0.8109i 1.1614 + 0.7909i 1.1559 + 0.7249i 0.3439 + 1.0689i 0.8555 + 1.0542i 0.8687 + 1.0407i 1.0335 + 0.8734i w23 −1.1516 + 0.7744i 1.2241 + 0.7367i 1.2163 + 0.6897i 0.3234 + 0.9962i 0.6363 + 1.2064i 0.6502 + 1.1951i 1.1905 + 0.6562i w24 −1.0113 + 0.2349i 0.9774 + 0.2034i 0.9601 + 0.1792i 0.1083 + 0.6241i 0.1088 + 0.9530i 0.0982 + 0.9745i 0.9948 + 0.1006i w25 −1.0113 + 0.2349i 0.9774 + 0.2034i 0.9601 + 0.1792i 0.1083 + 0.6241i 0.2464 + 0.9270i 0.2842 + 0.9344i 0.9492 + 0.2955i w26 −1.0113 + 0.2349i 0.9774 + 0.2034i 0.9601 + 0.1792i 0.1093 + 0.7170i 0.1124 + 1.1327i 0.1142 + 1.1448i 1.1596 + 0.1180i w27 −1.0113 + 0.2349i 0.9795 + 0.2417i 0.9601 + 0.1792i 0.1093 + 0.7170i 0.3160 + 1.0913i 0.3385 + 1.0973i 1.1091 + 0.3468i w28 −0.8869 + 0.5497i 0.8236 + 0.4847i 0.8069 + 0.4342i 0.3208 + 0.6318i 0.5707 + 0.7662i 0.6062 + 0.7465i 0.7476 + 0.6205i w29 −0.8869 + 0.5497i 0.8236 + 0.4847i 0.8069 + 0.4342i 0.3208 + 0.6318i 0.4490 + 0.8461i 0.4607 + 0.8538i 0.8645 + 0.4711i w30 −0.8869 + 0.5497i 0.8798 + 0.5374i 0.8601 + 0.4908i 0.3297 + 0.7141i 0.6961 + 0.8850i 0.7263 + 0.8764i 0.8777 + 0.7384i w31 −0.8869 + 0.5497i 0.8798 + 0.5374i 0.8601 + 0.4908i 0.3297 + 0.7141i 0.5229 + 1.0037i 0.5450 + 1.0067i 1.0128 + 0.5570i w32 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i 0.9688 + 0.1119i 0.3232 + 0.0858i 0.2660 + 0.0752i 0.0742 + 0.2556i w33 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i 0.9131 + 0.1709i 0.3232 + 0.0858i 0.2660 + 0.0752i 0.0742 + 0.2556i w34 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i 0.9688 + 0.1119i 0.4564 + 0.1187i 0.4544 + 0.0957i 0.0937 + 0.4473i w35 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1217 + 0.4285i 0.9131 + 0.1709i 0.4564 + 0.1187i 0.4544 + 0.0957i 0.0937 + 0.4473i w36 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i 0.2849 + 0.1302i 0.2572 + 0.1781i 0.2078 + 0.2475i w37 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i 0.2849 + 0.1302i 0.2572 + 0.1781i 0.2078 + 0.2475i w38 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i 0.3766 + 0.2539i 0.3660 + 0.2732i 0.2876 + 0.3664i w39 −0.1640 + 0.3056i 0.1509 + 0.3272i 0.1708 + 0.3974i 0.6419 + 0.1441i 0.3766 + 0.2539i 0.3660 + 0.2732i 0.2876 + 0.3664i w40 −0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i 1.3127 + 0.1240i 0.7502 + 0.1138i 0.7706 + 0.0922i 0.0813 + 0.7729i w41 −0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i 0.9572 + 0.4344i 0.7325 + 0.2088i 0.7407 + 0.2260i 0.2345 + 0.7412i w42 −0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i 1.2403 + 0.2631i 0.6473 + 0.1138i 0.6180 + 0.0927i 0.0756 + 0.6187i w43 −0.2071 + 0.6406i 0.1786 + 0.6836i 0.1744 + 0.7444i 1.0254 + 0.4130i 0.6339 + 0.1702i 0.6019 + 0.1658i 0.1815 + 0.5946i w44 −0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i 0.6207 + 0.4139i 0.5902 + 0.4857i 0.6007 + 0.4980i 0.5118 + 0.6039i w45 −0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i 0.6773 + 0.4284i 0.6355 + 0.4185i 0.6673 + 0.3928i 0.3821 + 0.6808i w46 −0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i 0.6207 + 0.4139i 0.5099 + 0.3795i 0.4786 + 0.3935i 0.4145 + 0.4734i w47 −0.3251 + 0.5895i 0.3585 + 0.6001i 0.3995 + 0.6028i 0.6207 + 0.4139i 0.5099 + 0.3795i 0.5176 + 0.3391i 0.3230 + 0.5340i w48 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i 0.0962 + 0.1443i 0.0755 + 0.1004i 0.0859 + 0.0728i w49 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i 0.0962 + 0.1443i 0.0755 + 0.1004i 0.0859 + 0.0728i w50 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i 0.0932 + 0.3972i 0.0822 + 0.4771i 0.5165 + 0.0795i w51 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.1249 + 0.1155i 0.0932 + 0.3972i 0.0822 + 0.4771i 0.5165 + 0.0795i w52 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i 0.0962 + 0.1443i 0.1016 + 0.2242i 0.2619 + 0.0919i w53 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i 0.0962 + 0.1443i 0.1016 + 0.2242i 0.2619 + 0.0919i w54 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i 0.1923 + 0.3624i 0.1916 + 0.3944i 0.4223 + 0.2036i w55 −0.3056 + 0.1640i 0.2707 + 0.1533i 0.2219 + 0.1422i 0.3743 + 0.1248i 0.1923 + 0.3624i 0.1916 + 0.3944i 0.4223 + 0.2036i w56 −0.6406 + 0.2071i 0.6459 + 0.1725i 0.6060 + 0.1399i 0.1166 + 0.3558i 0.1091 + 0.7656i 0.0930 + 0.8122i 0.8427 + 0.0848i w57 −0.6406 + 0.2071i 0.6459 + 0.1725i 0.6060 + 0.1399i 0.1166 + 0.3558i 0.1699 + 0.7537i 0.2215 + 0.7840i 0.8036 + 0.2443i w58 −0.6406 + 0.2071i 0.6459 + 0.1725i 0.6060 + 0.1399i 0.1166 + 0.3558i 0.1142 + 0.5964i 0.0937 + 0.6514i 0.6933 + 0.0788i w59 −0.6406 + 0.2071i 0.6459 + 0.1725i 0.6060 + 0.1399i 0.1166 + 0.3558i 0.1142 + 0.5964i 0.1540 + 0.6366i 0.6649 + 0.1793i w60 −0.5895 + 0.3251i 0.5863 + 0.3220i 0.5660 + 0.3001i 0.3519 + 0.3779i 0.4294 + 0.6363i 0.4810 + 0.6306i 0.6321 + 0.5048i w61 −0.5895 + 0.3251i 0.5863 + 0.3220i 0.5660 + 0.3001i 0.3519 + 0.3779i 0.3744 + 0.6744i 0.3856 + 0.7037i 0.7279 + 0.3892i w62 −0.5895 + 0.3251i 0.5863 + 0.3220i 0.5660 + 0.3001i 0.3519 + 0.3779i 0.3120 + 0.5292i 0.3527 + 0.5230i 0.5265 + 0.3752i w63 −0.5895 + 0.3251i 0.5863 + 0.3220i 0.5660 + 0.3001i 0.3519 + 0.3779i 0.3120 + 0.5292i 0.3100 + 0.5559i 0.5885 + 0.3076i D3) 1024-ConQQAM NUC

w/code rate 5/15 6/15 7/15 8/15 9/15 w1 1.9734 + 0.6475i 1.3555 + 1.1873i 0.1119 + 1.7802i 0.0966 + 1.7257i 0.1003 + 1.7258i w1 1.7794 + 0.9784i 1.0871 + 1.4324i 0.2324 + 2.0028i 0.1799 + 1.9540i 0.1722 + 1.9393i w2 0.4642 + 1.8997i 1.5003 + 1.4954i 0.1416 + 1.5629i 0.1216 + 1.5846i 0.0956 + 1.5505i w3 0.8851 + 1.9829i 1.1127 + 1.7206i 0.1594 + 1.5242i 0.2925 + 1.5646i 0.2621 + 1.5384i w4 0.2393 + 1.3953i 0.2080 + 2.0408i 0.8323 + 1.8356i 0.7016 + 1.7899i 0.6393 + 1.7656i w5 0.2393 + 1.3953i 0.7958 + 1.4948i 0.5102 + 1.8340i 0.4407 + 1.8294i 0.3969 + 1.7846i w6 0.2534 + 1.4887i 0.5864 + 1.9337i 0.4040 + 1.5212i 0.6347 + 1.5255i 0.6050 + 1.5226i w7 0.2393 + 1.3953i 0.7534 + 1.6241i 0.4040 + 1.5212i 0.4512 + 1.5701i 0.4221 + 1.5525i w8 2.0528 + 0.2197i 1.0007 + 1.0906i 0.1151 + 1.2407i 0.0862 + 1.3505i 0.0799 + 1.3199i w9 1.6003 + 1.3185i 0.9660 + 1.1761i 0.1151 + 1.2407i 0.2378 + 1.3237i 0.2425 + 1.3020i w10 0.1805 + 2.0047i 0.9193 + 1.0471i 0.1219 + 1.3013i 0.0908 + 1.3996i 0.0833 + 1.3795i w11 1.3579 + 1.6516i 0.8849 + 1.0953i 0.1219 + 1.3013i 0.2515 + 1.3739i 0.2487 + 1.3670i w12 0.1534 + 1.4308i 0.8849 + 1.0953i 0.3445 + 1.2020i 0.5334 + 1.2483i 0.5352 + 1.2272i w13 0.1532 + 1.3846i 0.8468 + 1.2089i 0.3445 + 1.2020i 0.4112 + 1.2843i 0.3974 + 1.2734i w14 0.1492 + 1.5422i 0.8508 + 1.0556i 0.3544 + 1.2507i 0.5634 + 1.3178i 0.5666 + 1.3364i w15 0.1534 + 1.4308i 0.8263 + 1.1474i 0.3544 + 1.2507i 0.4201 + 1.3450i 0.4117 + 1.3592i w16 0.7168 + 1.4422i 1.3097 + 0.8498i 1.0272 + 1.4142i 1.3853 + 1.1272i 1.4270 + 1.2101i w17 0.7660 + 1.4662i 1.2703 + 0.7763i 0.9293 + 1.2912i 1.3369 + 1.3509i 1.2720 + 1.3913i w18 0.7079 + 1.5478i 1.2076 + 0.7137i 1.0481 + 1.1977i 1.2114 + 1.0355i 1.2297 + 1.0783i w19 0.7980 + 1.5832i 1.2076 + 0.7137i 1.0032 + 1.1946i 1.1082 + 1.1877i 1.1069 + 1.2225i w20 0.5597 + 1.2994i 1.0228 + 0.5992i 0.8044 + 1.5410i 0.9388 + 1.6673i 0.8723 + 1.6736i w21 0.5597 + 1.2994i 1.0228 + 0.5992i 0.7504 + 1.3903i 1.1497 + 1.5199i 1.0846 + 1.5470i w22 0.5597 + 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0.3109i 0.5929 + 0.0575i 0.0601 + 0.6048i 0.6363 + 0.0929i w234 0.0568 + 0.3109i 0.6701 + 0.0660i 0.0373 + 0.6868i 0.7212 + 0.0348i w235 0.0568 + 0.3109i 0.6701 + 0.0660i 0.0991 + 0.6806i 0.7163 + 0.1032i w236 0.1690 + 0.3113i 0.5701 + 0.1700i 0.1750 + 0.5791i 0.6122 + 0.2069i w237 0.1690 + 0.3113i 0.5701 + 0.1700i 0.1750 + 0.5791i 0.6268 + 0.1493i w238 0.1690 + 0.3113i 0.6451 + 0.1940i 0.2285 + 0.6480i 0.6853 + 0.2349i w239 0.1690 + 0.3113i 0.6451 + 0.1940i 0.1719 + 0.6651i 0.7046 + 0.1699i w240 0.4100 + 0.4451i 0.6189 + 0.5514i 0.5654 + 0.6265i 0.6521 + 0.5790i w241 0.4100 + 0.4451i 0.6665 + 0.4960i 0.5027 + 0.6784i 0.7094 + 0.5138i w242 0.4100 + 0.4451i 0.5667 + 0.4955i 0.5121 + 0.5697i 0.6013 + 0.5265i w243 0.4100 + 0.4451i 0.6028 + 0.4546i 0.4555 + 0.6162i 0.6524 + 0.4667i w244 0.2871 + 0.4459i 0.7506 + 0.3592i 0.3624 + 0.7630i 0.7984 + 0.3673i w245 0.2871 + 0.4459i 0.7156 + 0.4239i 0.4347 + 0.7245i 0.7580 + 0.4428i w246 0.2871 + 0.4459i 0.6809 + 0.3299i 0.3286 + 0.6923i 0.7298 + 0.3325i w247 0.2871 + 0.4459i 0.6544 + 0.3786i 0.3924 + 0.6581i 0.6964 + 0.4016i w243 0.4066 + 0.3107i 0.4637 + 0.3699i 0.3814 + 0.4708i 0.5005 + 0.4181i w249 0.4066 + 0.3107i 0.4637 + 0.3699i 0.3814 + 0.4708i 0.5359 + 0.3697i w250 0.4066 + 0.3107i 0.5223 + 0.4197i 0.4563 + 0.5135i 0.5509 + 0.4732i w251 0.4066 + 0.3107i 0.5223 + 0.4197i 0.4093 + 0.5514i 0.5953 + 0.4196i w252 0.2858 + 0.3115i 0.5265 + 0.2736i 0.2826 + 0.5338i 0.5896 + 0.2686i w253 0.2858 + 0.3115i 0.5265 + 0.2736i 0.2826 + 0.5338i 0.5635 + 0.3210i w254 0.2858 + 0.3115i 0.5946 + 0.3133i 0.2962 + 0.6193i 0.6601 + 0.3001i w255 0.2858 + 0.3115i 0.5946 + 0.3133i 0.3481 + 0.5915i 0.6312 + 0.3619i

For further illustration FIGS. 11A-11C show two of the proposed 1024-N²-NUCs optimized for 27 dB SNR (FIG. 11A) and 8 dB SNR (FIG. 11B) and an enlargement of the N²-NUC optimized for 8 dB SNR (FIG. 11C), illustrating that several constellation points tend to merge at low SNR.

FIGS. 12A and 12B show two 64-QQAMs optimized for 10 dB SNR (FIG. 12A) and 15 dB SNR (FIG. 12B). In contrast to the rectangular shaped N²-NUCs, the QQAM constellations have a circular shape, reducing the peak-power of the constellation compared to N²-NUCs, by avoiding the constellation points near the corner. The constellation points of large QQAMs tend to merge in the same way like N²-NUCs when being optimized at low SNR. This is e.g. the case in the inner circle of the 64-QQAM optimized at 10 dB target SNR shown in FIG. 12A.

The condensation of constellations, especially if optimized for very low SNR, may sometimes lead to (complete) puncturing of the least significant bits of the constellation. This is for example the case for a 256-N²-NUC optimized for 0 dB SNR, which results in a QPSK constellation with all 64 constellation points of one quadrant having exactly the same constellation point position. When demapping such a constellation in the receiver, the first two bits can be restored by means of the four different constellation point positions. In case of an unencoded system the remaining 6 least significant bits could not be restored. This is however possible using a BICM chain with state-of-the-art forward error correction codes, which is able to correct the remaining bits based on the information of the two most significant bits. Such a BICM chain, as e.g. conventionally used in systems according to various DVB standards, is thus preferably used in the transmitter and the receiver of a communication system according to the present disclosure. Preferably, the use of such a BICM chain is assumed to be used when performing an optimization of the constellations, and the BICM capacity is the target capacity during the optimization process. In the extreme example mentioned above, it would alternatively possible to directly transmit a QPSK constellation carrying only 2 bits per constellation symbol to avoid the increased demapping complexity. For higher SNR the transmission of condensed non-uniform constellations with very high order is however advantageous from a performance perspective compared to smaller non-uniform constellations.

In still another embodiment the modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation obtained from a constellation from anyone of groups A, B, C or D through rotation by an angle around the origin. In other words, one or more of the following “invariant transformations” do not affect the properties of a mapping:

1. rotation of all symbols by an arbitrary angle φ,

2. inversion of m-th bit y_m=b∈{0,1} to y_m= b, where the bar indicates inversion,

3. interchanging of bit positions y_k1 and y_k2,

4. reflection on Re{x1}- and/or Im{x1}-axis.

In still another embodiment the modulator of the disclosed coding and modulation apparatus modulates said cell words into constellation values of a non-uniform constellation wherein said modulator is configured to use, based on the total number M of constellation points of the constellation and the code rate, a non-uniform constellation from a group E of constellations comprising one or more of the following constellations defined by the constellation position vector w_(0 . . . M). Such constellations can not be described by the symmetry from the QQAM in a straightforward manner. Therefore, the complete constellation position vector with M entries will be used.

The following non-uniform constellations comprised in group E are proposed:

E) M-QAM non-uniform constellations of group E:

E1) 8-QAM 2D NUC

code rate w 100/180 or 104/180 23/36 25/36 or 13/18 32/45 (16200 length code) w0 −0.2330 −0.2231 −0.2416 −0.2416 + 0.0000i w1 −0.5414 − 0.8712i −0.5520 − 0.8882i −0.5547 − 0.8925i −0.5547 − 0.8925i w2 −0.5414 + 0.8712i −0.5520 + 0.8882i −0.5547 + 0.8925i −0.5547 + 0.8925i w3 −1.3570 −1.3279 −1.3165 −1.3165 + 0.0000i w4  0.2330  0.2231  0.2416 0.2416 w5  0.5414 − 0.8712i  0.5520 − 0.8882i  0.5547 − 0.8925i  0.5547 − 0.8925i w6  0.5414 + 0.8712i  0.5520 + 0.8882i  0.5547 + 0.8925i  0.5547 + 0.8925i w7  1.3570  1.3279  1.3165 1.3165 code rate w 19/30 w0  0.8753 + 0.5486i w1 +1.3470i w2 +0.2266i w3 −0.8753 + 0.5486i w4  0.8753 − 0.5486i w5 −1.3470i w6 −0.2266i w7 −0.8753 − 0.5486i E2) 16-QAM 2D NUC

90/180 or 96/180 w/code or 100/180 26/45 3/5 28/45 or 23/36 w0  0.6186 + 0.2544i  0.4718 + 0.2606i  0.4718 + 0.2606i  0.4936 + 0.2530i w1 −0.6186 + 0.2544i −0.4718 + 0.2606i −0.4718 + 0.2606i −0.4936 + 0.2530i w2  0.6186 − 0.25441  0.4718 − 0.2606i  0.4718 − 0.2606i  0.4936 − 0.2530i w3 −0.6186 − 0.2544i −0.4718 − 0.2606i −0.4718 − 0.2606i −0.4936 − 0.2530i w4  1.2080 + 0.5377i  1.2088 + 0.4984i  0.2606 + 0.4718i  0.2530 + 0.4936i w5 −1.2080 + 0.5377i −1.2088 + 0.4984i −0.2606 + 0.4718i −0.2530 + 0.4936i w6  1.2080 − 0.5377i  1.2088 − 0.4984i  0.2606 − 0.4718i  0.2530 − 0.4936i w7 −1.2080 − 0.5377i −1.2088 − 0.4984i −0.2606 − 0.4718; −0.2530 − 0.4936i w8  0.2213 + 0.4416i  0.2606 + 0.4718i  1.2088 + 0.4984i  1.2040 + 0.4925i w9 −0.2213 + 0.4416i −0.2606 + 0.4718i −1.2088 + 0.4984i −1.2040 + 0.4925i w10  0.2213 − 0.4416i  0.2606 − 0.4718i  1.2088 − 0.4984i  1.2040 − 0.4925i w11 −0.2213 − 0.4416i −0.2606 − 0.4718i −1.2088 − 0.4984i −1.2040 − 0.4925i w12  0.4487 + 1.1657i  0.4984 + 1.2088i  0.4984 + 1.2088i  0.4925 + 1.2040i w13 −0.4487 + 1.1657i −0.4984 + 1.2088i −0.4984 + 1.2088i −0.4925 + 1.2040i w14  0.4487 − 1.1657i  0.4984 − 1.2088i  0.4984 − 1.2088i  0.4925 − 1.2040i w15 −0.4487 − 1.1657i −0.4984 − 1.2088i −0.4984 − 1.2088i −0.4925 − 1.2040i w/code 25/36 or 13/18 140/180 or 154/180 20/30 w0  0.6186 + 0.2544i  0.6578 + 0.2571i  0.5061 + 0.2474i w1 −0.6186 + 0.2544i −0.6578 + 0.2571i  0.2474 + 0.5061i w2  0.6186 − 0.2544i  0.6578 − 0.2571i −0.5061 + 0.2474i w3 −0.6186 − 0.2544i −0.6578 − 0.2571i −0.2474 + 0.5061i w4  1.2080 + 0.5377i  1.2088 + 0.5659i  0.5061 − 0.2474i w5 −1.2080 + 0.5377i −1.2088 + 0.5659i  0.2474 − 0.5061i w6  1.2080 − 0.5377i  1.2088 − 0.5659i −0.5061 − 0.2474i w7 −1.2080 − 0.5377i −1.2088 − 0.5659i −0.2474 − 0.5061i w8  0.2213 + 0.4416i  0.2173 + 0.4189i  1.2007 + 0.4909i w9 −0.2213 + 0.4416i −0.2173 + 0.4189i  0.4909 + 1.2007i w10  0.2213 − 0.4416i  0.2173 − 0.4189i −1.2007 + 0.4909i w11 −0.2213 − 0.4416i −0.2173 − 0.4189i −0.4909 + 1.2007i w12  0.4487 + 1.1657i  0.4326 + 1.1445i  1.2007 − 0.4909i w13 −0.4487 + 1.1657i −0.4326 + 1.1445i  0.4909 − 1.2007i w14  0.4487 − 1.1657i  0.4326 − 1.1445i −1.2007 − 0.4909i w15 −0.4487 − 1.1657i −0.4326 − 1.1445i −0.4909 − 1.2007i E3) 32-QAM 2D NUC

code rate w 2/3 or 25/36 128/180 or 132/180 or 140/180 w0  0.1929 + 0.1744i  0.2121 + 0.1569i w1  0.2283 + 0.5036i −0.2121 + 0.1569i w2  0.1929 − 0.1744i  0.2121 − 0.1569i w3  0.2283 − 0.5036i −0.2121 − 0.1559i w4 −0.1929 + 0.1744i  0.7185 + 0.1739i w5 −0.2283 + 0.5036i −0.7185 + 0.1739i w6 −0.1929 − 0.1744i  0.7185 − 0.1739i w7 −0.2283 − 0.5036i −0.7185 − 0.1739i w8  0.3541 + 1.4168i  0.3478 + 1.4027i w9  0.2627 + 0.9170i −0.3478 + 1.4027i w10  0.3541 − 1.4168i  0.3478 − 1.4027i w11  0.2627 − 0.9170i −0.3478 − 1.4027i w12 −0.3541 + 1.4168i  1.2867 + 0.3209i w13 −0.2627 + 0.9170i −1.2867 + 0.3209i w14 −0.3541 − 1.4168i  1.2867 − 0.3209i w15 −0.2627 − 0.9170i −1.2867 − 0.3209i w16  0.6963 + 0.1782i  0.2258 + 0.5089i w17  0.6364 + 0.4437i −0.2258 + 0.5089i w18  0.6963 − 0.1782i  0.2258 − 0.5089i w19  0.6364 − 0.4437i −0.2258 − 0.5089i w20 −0.6963 + 0.1782i  0.6431 + 0.5018i w21 −0.6364 + 0.4437i −0.6431 + 0.5018i w22 −0.6963 − 0.1782i  0.6431 − 0.5018i w23 −0.6364 − 0.4437i −0.6431 − 0.5018i w24  1.3162 + 0.3270i  0.2443 + 0.9172i w25  0.9382 + 0.8637i −0.2443 + 0.9172i w26  1.3162 − 0.3270i  0.2443 − 0.9172i w27  0.9382 − 0.8637i −0.2443 − 0.9172i w28 −1.3162 + 0.3270i  0.9274 + 0.8949i w29 −0.9382 + 0.8637i −0.9274 + 0.8949i w30 −1.3162 − 0.3270i  0.9274 − 0.8949i w31 −0.9382 − 0.8637i −0.9274 − 0.8949i E4) 64-QAM U2D NUC

code rate w 132/180 140/180 or 144/180 7/9 or 4/5 150/180 5/6 w0  0.2878 + 1.4388i  0.9856 − 0.4661i  0.2775 + 1.4188i  1.0161 − 0.4912i  1.0161 − 0.4912i w1 −0.2878 + 1.4388i −0.9856 − 0.4661i  0.1138 + 0.3999i  1.0161 + 0.4912i  0.1287 − 0.4061i w2  0.2878 − 1.4388i  0.9856 + 0.4661i  0.2177 + 1.0243i −1.0161 − 0.4912i  1.0441 − 0.1581i w3 −0.2878 − 1.4388i −0.9856 + 0.4661i  0.1487 + 0.7260i −1.0161 + 0.4912i  0.1321 − 0.1317i w4  0.1177 + 0.4119i  1.1580 − 0.8178i  0.7921 + 1.2096i  0.1287 − 0.4061i  0.6966 − 0.4427i w5 −0.1177 + 0.4119i −1.1580 − 0.8178i  0.2891 + 0.3910i  0.1287 + 0.4061i  0.4025 − 0.4142i w6  0.1177 − 0.4119i  1.1580 + 0.8178i  0.6056 + 0.8481i −0.1287 − 0.4061i  0.6995 − 0.1411i w7 −0.1177 − 0.4119i −1.1580 + 0.8178i  0.4397 + 0.5853i −0.1287 + 0.4061i  0.4035 − 0.1354i w8  1.4656 + 0.2931i  0.1264 − 0.4145i  0.2775 − 1.4188i  1.1306 − 0.8649i  1.0161 + 0.4912i w9 −1.4656 + 0.2931i −0.1264 − 0.4145i  0.1138 − 0.3999i  1.1306 + 0.8649i  0.1287 + 0.4061i w10  1.4656 − 0.2931i  0.1264 + 0.4145i  0.2177 − 1.0243i −1.1306 − 0.8649i  1.0441 + 0.1581i w11 −1.4656 − 0.2931i −0.1264 + 0.4145i  0.1487 − 0.7260i −1.1306 + 0.8649i  0.1321 + 0.1317i w12  0.1678 + 0.1166i  0.1416 − 0.7330i  0.7921 − 1.2096i  0.1385 − 0.7199i  0.6966 + 0.4427i w13 −0.1678 + 0.1166i −0.1416 − 0.7330i  0.2891 − 0.3910i  0.1385 + 0.7199i  0.4025 + 0.4142i w14  0.1678 − 0.1166i  0.1416 + 0.7330i  0.6056 − 0.8481i −0.1385 − 0.7199i  0.6995 + 0.1411i w15 −0.1678 − 0.1166i −0.1416 + 0.7330i  0.4397 − 0.5853i −0.1385 + 0.7199i  0.4035 + 0.1354i w16  0.2219 + 1.0386i  1.0366 − 0.1534i  1.4730 + 0.3019i  1.0441 − 0.1581i  1.1306 − 0.8649i w17 −0.2219 + 1.0386i −1.0366 − 0.1534i  0.1419 + 0.1122i  1.0441 + 0.1581i  0.1385 − 0.7199i w18  0.2219 − 1.0386i  1.0366 + 0.1534i  1.0895 + 0.2172i −1.0441 − 0.1581i  1.4516 − 0.2578i w19 −0.2219 − 1.0386i −1.0366 + 0.1534i  0.7863 + 0.1337i −1.0441 + 0.1581i  0.1689 − 1.0567i w20  0.1559 + 0.7442i  1.4529 − 0.2702i  1.2124 + 0.8333i  0.1321 − 0.1317i  0.6874 − 0.8123i w21 −0.1559 + 0.7442i −1.4529 − 0.2702i  0.3733 + 0.1498i  0.1321 + 0.1317i  0.4017 − 0.7107i w22  0.1559 − 0.7442i  1.4529 + 0.2702i  0.8988 + 0.5768i −0.1321 − 0.1317i  0.6750 − 1.2072i w23 −0.1559 − 0.7442i −1.4529 + 0.2702i  0.6394 + 0.3211i −0.1321 + 0.1317i  0.2558 − 1.4247i w24  1.0649 + 0.2069i  0.1272 − 0.1353i  1.4730 − 0.3019i  1.4516 − 0.2578i  1.1306 + 0.8649i w25 −1.0649 + 0.2069i −0.1272 − 0.1353i  0.1419 − 0.1122i  1.4516 + 0.2578i  0.1385 + 0.7199i w26  1.0649 − 0.2069i  0.1272 + 0.1353i  1.0895 − 0.2172i −1.4516 − 0.2578i  1.4516 + 0.2578i w27 −1.0649 − 0.2069i −0.1272 + 0.1353i  0.7863 − 0.1337i −1.4516 + 0.2578i  0.1689 + 1.0567i w28  0.7408 + 0.1355i  0.1686 − 1.0718i  1.2124 − 0.8333i  0.1689 − 1.0567i  0.6874 + 0.8123i w29 −0.7408 + 0.1355i −0.1686 − 1.0718i  0.3733 − 0.1498i  0.1689 + 1.0567i  0.4017 + 0.7107i w30  0.7408 − 0.1355i  0.1686 + 1.0718i  0.8988 − 0.5768i −0.1689 − 1.0567i  0.6750 + 1.2072i w31 −0.7408 − 0.1355i −0.1686 + 1.0718i  0.6394 − 0.3211i −0.1689 + 1.0567i  0.2558 + 1.4247i w32  0.8133 + 1.2150i  0.6825 − 0.4329i −0.2775 + 1.4188i  0.6966 − 0.4427i −1.0161 − 0.4912i w33 −0.8133 + 1.2150i −0.6825 − 0.4329i −0.1138 + 0.3999i  0.6966 + 0.4427i −0.1287 − 0.4061i w34  0.8133 − 1.2150i  0.6825 + 0.4329i −0.2177 + 1.0243i −0.6966 − 0.4427i −1.0441 − 0.1581i w35 −0.8133 − 1.2150i −0.6825 + 0.4329i −0.1487 + 0.7260i −0.6966 + 0.4427i −0.1321 − 0.1317i w36  0.2516 + 0.3998i  0.6913 − 0.8132i −0.7921 + 1.2096i  0.4025 − 0.4142i −0.6966 − 0.4427i w37 −0.2516 + 0.3998i −0.6913 − 0.8132i −0.2891 + 0.3910i  0.4025 + 0.4142i −0.4025 − 0.4142i w38  0.2516 − 0.3998i  0.6913 + 0.8132i −0.6056 + 0.8481i −0.4025 − 0.4142i −0.6995 − 0.1411i w39 −0.2516 − 0.3998i −0.6913 + 0.8132i −0.4397 + 0.5853i −0.4025 + 0.4142i −0.4035 − 0.1354i w40  1.2278 + 0.8230i  0.3948 − 0.4179i −0.2775 − 1.4188i  0.6874 − 0.8123i −1.0161 + 0.4912i w41 −1.2278 + 0.8230i −0.3948 − 0.4179i −0.1138 − 0.3999i  0.6874 + 0.8123i −0.1287 + 0.4061i w42  1.2278 − 0.8230i  0.3948 + 0.4179i −0.2177 − 1.0243i −0.6874 − 0.8123i −1.0441 + 0.1581i w43 −1.2278 − 0.8230i −0.3948 + 0.4179i −0.1487 − 0.7260i −0.6874 + 0.8123i −0.1321 + 0.1317i w44  0.3325 + 0.1582i  0.4018 − 0.7177i −0.7921 − 1.2096i  0.4017 − 0.7107i −0.6966 + 0.4427i w45 −0.3325 + 0.1582i −0.4018 − 0.7177i −0.2891 − 0.3910i  0.4017 + 0.7107i −0.4025 + 0.4142i w46  0.3325 − 0.1582i  0.4018 + 0.7177i −0.6056 − 0.8481i −0.4017 − 0.7107i −0.6995 + 0.1411i w47 −0.3325 − 0.1582i −0.4018 + 0.7177i −0.4397 − 0.5853i −0.4017 + 0.7107i −0.4035 + 0.1354i w48  0.6145 + 0.8494i  0.6796 − 0.1340i −1.4730 + 0.3019i  0.6995 − 0.1411i −1.1306 − 0.8649i w49 −0.6145 + 0.8494i −0.6796 − 0.1340i −0.1419 + 0.1122i  0.6995 + 0.1411i −0.1385 − 0.7199i w50  0.6145 − 0.8494i  0.6796 + 0.1340i −1.0895 + 0.2172i −0.6995 − 0.1411i −1.4516 − 0.2578i w51 −0.6145 − 0.8494i −0.6796 + 0.1340i −0.7863 + 0.1337i −0.6995 + 0.1411i −0.1689 − 1.0567i w52  0.4328 + 0.5954i  0.7097 − 1.2125i −1.2124 + 0.8333i  0.4035 − 0.1354i −0.6874 − 0.8123i w53 −0.4328 + 0.5954i −0.7097 − 1.2125i −0.3733 + 0.1498i  0.4035 + 0.1354i −0.4017 − 0.7107i w54  0.4328 − 0.5954i  0.7097 + 1.2125i −0.8988 + 0.5768i −0.4035 − 0.1354i −0.6750 − 1.2072i w55 −0.4328 − 0.5954i −0.7097 + 1.2125i −0.6394 + 0.3211i −0.4035 + 0.1354i −0.2558 − 1.4247i w56  0.8971 + 0.5677i  0.3877 − 0.1359i −1.4730 − 0.3019i  0.6750 − 1.2072i −1.1306 + 0.8649i w57 −0.8971 + 0.5677i −0.3877 − 0.1359i −0.1419 − 0.1122i  0.6750 + 1.2072i −0.1385 + 0.7199i w58  0.8971 − 0.5677i  0.3877 + 0.1359i −1.0895 − 0.2172i −0.6750 − 1.2072i −1.4516 + 0.2578i w59 −0.8971 − 0.5677i −0.3877 + 0.1359i −0.7863 − 0.1337i −0.6750 + 1.2072i −0.1689 + 1.0567i w60  0.6200 + 0.3227i  0.2732 − 1.4375i −1.2124 − 0.8333i  0.2558 − 1.4247i −0.6874 + 0.8123i w61 −0.6200 + 0.3227i −0.2732 − 1.4375i −0.3733 − 0.1498i  0.2558 + 1.4247i −0.4017 + 0.7107i w62  0.6200 − 0.3227i  0.2732 + 1.4375i −0.8988 − 0.5768i −0.2558 − 1.4247i −0.6750 + 1.2072i w63 −0.6200 − 0.3227i −0.2732 + 1.4375i −0.6394 − 0.3211i −0.2558 + 1.4247i −0.2558 + 1.4247i code rate w 21/30 23/30 w0  0.2994 + 1.4627i  0.2878 + 1.4388i w1  0.8421 + 1.2279i  0.8133 + 1.2150i w2  1.4625 + 0.2831i  0.2878 − 1.4388i w3  1.2488 + 0.8125i  0.8133 − 1.2150i w4  0.1299 + 0.4106i −0.2878 + 1.4388i w5  0.1823 + 0.3822i −0.8133 + 1.2150i w6  0.2512 + 0.1287i −0.2878 − 1.4388i w7  0.2598 + 0.1538i −0.8133 − 1.2150i w8 −0.2994 + 1.4627i  0.2219 + 1.0386i w9 −0.8421 + 1.2279i  0.6145 + 0.8494i w10 −1.4625 + 0.2831i  0.2219 − 1.0386i w11 −1.2488 + 0.8125i  0.6145 − 0.8494i w12 −0.1299 + 0.4106i −0.2219 + 1.0386i w13 −0.1823 + 0.3822i −0.6145 + 0.8494i w14 −0.2512 + 0.1287i −0.2219 − 1.0386i w15 −0.2598 + 0.1538i −0.6145 − 0.8494i w16  0.2994 − 1.4627i  0.1177 + 0.4119i w17  0.8421 − 1.2279i  0.2516 + 0.3998i w18  1.4625 − 0.2831i  0.1177 − 0.4119i w19  1.2488 − 0.8125i  0.2516 − 0.3998i w20  0.1299 − 0.4106i −0.1177 + 0.4119i w21  0.1823 − 0.3822i −0.2516 + 0.3998i w22  0.2512 − 0.1287i −0.1177 − 0.4119i w23  0.2598 − 0.1538i −0.2516 − 0.3998i w24 −0.2994 − 1.4627i  0.1559 + 0.7442i w25 −0.8421 − 1.2279i  0.4328 + 0.5954i w26 −1.4625 − 0.2831i  0.1559 − 0.7442i w27 −1.2488 − 0.8125i  0.4328 − 0.5954i w28 −0.1299 − 0.4106i −0.1559 + 0.7442i w29 −0.1823 − 0.3822i −0.4328 + 0.5954i w30 −0.2512 − 0.1287i −0.1559 − 0.7442i w31 −0.2598 − 0.1538i −0.4328 − 0.5954i w32  0.2187 + 1.0467i  1.4656 + 0.2931i w33  0.6179 + 0.8606i  1.2278 + 0.8230i w34  1.0296 + 0.1935i  1.4656 − 0.2931i w35  0.8910 + 0.5575i  1.2278 − 0.8230i w36  0.1749 + 0.7440i −1.4656 + 0.2931i w37  0.4168 + 0.6160i −1.2278 + 0.8230i w38  0.6759 + 0.1488i −1.4656 − 0.2931i w39  0.6111 + 0.3494i −1.2278 − 0.8230i w40 −0.2187 + 1.0467i  1.0649 + 0.2069i w41 −0.6179 + 0.8606i  0.8971 + 0.5677i w42 −1.0296 + 0.1935i  1.0649 − 0.2069i w43 −0.8910 + 0.5575i  0.8971 − 0.5677i w44 −0.1749 + 0.7440i −1.0649 + 0.2069i w45 −0.4168 + 0.6160i −0.8971 + 0.5677i w46 −0.6759 + 0.1488i −1.0649 − 0.2069i w47 −0.6111 + 0.3494i −0.8971 − 0.5677i w48  0.2187 − 1.0467i  0.1678 + 0.1166i w49  0.6179 − 0.8606i  0.3325 + 0.1582i w50  1.0296 − 0.1935i  0.1678 − 0.1166i w51  0.8910 − 0.5575i  0.3325 − 0.1582i w52  0.1749 − 0.7440i −0.1678 + 0.1166i w53  0.4168 − 0.6160i −0.3325 + 0.1582i w54  0.6759 − 0.1488i −0.1678 − 0.1166i w55  0.6111 − 0.3494i −0.3325 − 0.1582i w56 −0.2187 − 1.0467i  0.7408 + 0.1355i w57 −0.6179 − 0.8606i  0.6200 + 0.3227i w58 −1.0296 − 01935i  0.7408 − 0.1355i w59 −0.8910 − 0.5575i  0.6200 − 0.3227i w60 −0.1749 − 0.7440i −0.7408 + 0.1355i w61 −0.4168 − 0.6160i −0.6200 + 0.3227i w62 −0.6759 − 0.1488i −0.7408 − 0.1355i w63 −0.6111 − 0.3494i −0.6200 − 0.3227i E5) 128-QAM 2D NUC

code rate w 135/180 140/180 w0  1.1438 + 0.4323i  1.1021 + 0.4008i w1  0.9006 + 0.4173i  0.8557 + 0.4063i w2  0.5578 + 0.4216i  0.5499 + 0.4932i w3  0.6797 + 0.3850i  0.6209 + 0.3666i w4  1.1837 + 0.8001i  0.1083 + 0.5383i w5  0.9008 + 0.6972i  0.1008 + 0.3768i w6  0.5354 + 0.6351i  0.3317 + 0.5081i w7  0.6549 + 0.7546i  0.3204 + 0.3654i w8  1.1598 + 0.1437i  1.0961 + 0.1322i w9  0.9102 + 0.1324i  0.8513 + 0.1298i w10  0.5755 + 0.1222i  0.5343 + 0.0862i w11  0.6787 + 0.1401i  0.6260 + 0.1694i w12  1.4855 + 0.1793i  0.1082 + 0.0775i w13  1.5681 + 0.5636i  0.1037 + 0.2076i w14  0.9727 + 1.2109i  0.3325 + 0.0910i w15  0.7102 + 1.0171i  0.3189 + 0.2137i w16  0.1045 + 0.4578i  1.1805 + 0.7084i w17  0.1067 + 0.3515i  0.8930 + 0.6876i w18  0.3411 + 0.4189i  0.5622 + 0.6728i w19  0.3088 + 0.3456i  0.6763 + 0.8371i w20  0.1121 + 0.6749i  0.1170 + 0.7305i w21  0.1179 + 0.8955i  0.1259 + 0.9453i w22  0.3342 + 0.6628i  0.3446 + 0.71295 w23  0.3585 + 0.8699i  0.3857 + 0.9233i w24  0.1075 + 0.0988i  1.4038 + 0.1588i w25  0.1136 + 0.1402i  1.5205 + 0.4861i w26  0.3700 + 0.1151i  1.0743 + 1.1040i w27  0.3294 + 0.1448i  0.7513 + 1.0912i w28  0.1933 + 1.4991i  0.1870 + 1.5043i w29  0.1440 + 1.1691i  0.1454 + 1.1973i w30  0.5835 + 1.4250i  0.5736 + 1.4545i w31  0.4344 + 1.1140i  0.4433 + 1.1606i w32 −1.1438 + 0.4323i  1.1021 − 0.4008i w33 −0.9006 + 0.4173i  0.8567 − 0.4063i w34 −0.5578 + 0.4216i  0.5499 − 0.4932i w35 −0.6797 + 0.3850i  0.6209 − 0.3666i w36 −1.1837 + 0.8001i  0.1083 − 0.5383i w37 −0.9008 + 0.6972i  0.1008 − 0.3768i w38 −0.5354 + 0.6351i  0.3317 − 0.5081i w39 −0.6549 + 0.7546i  0.3204 − 0.3654i w40 −1.1598 + 0.1437i  1.0961 − 0.1322i w41 −0.9102 + 0.1324i  0.8513 − 0.1298i w42 −0.5755 + 0.1222i  0.5343 − 0.0862i w43 −0.6787 + 0.1401i  0.6260 − 0.1694i w44 −1.4855 + 0.1793i  0.1082 − 0.0775i w45 −1.5681 + 0.5636i  0.1037 − 0.2076i w46 −0.9727 + 1.2109i  0.3325 − 0.0910i w47 −0.7102 + 1.0171i  0.3189 − 0.2137i w48 −0.1045 + 0.4578i  1.1805 − 0.7084i w49 −0.1067 + 0.3515i  0.8930 − 0.6876i w50 −0.3411 + 0.4189i  0.5622 − 0.6728i w51 −0.3088 + 0.3456i  0.6763 − 0.8371i w52 −0.1121 + 0.6749i  0.1170 − 0.7305i w53 −0.1179 + 0.8955i  0.1259 − 0.9453i w54 −0.3342 + 0.6628i  0.3446 − 0.7129i w55 −0.3585 + 0.8699i  0.3857 − 0.9233i w56 −0.1075 + 0.0988i  1.4038 − 0.1588i w57 −0.1136 + 0.1402i  1.5205 − 0.4861i w58 −0.3700 + 0.1151i  1.0743 − 1.1040i w59 −0.3294 + 0.1448i  0.7513 − 1.0912i w60 −0.1933 + 1.4991i  0.1870 − 1.5043i w61 −0.1440 + 1.1691i  0.1454 − 1.1973i w62 −0.5835 + 1.4250i  0.5736 − 1.4545i w63 −0.4344 + 1.1140i  0.4433 − 1.1606i w64  1.1438 − 0.4323i −1.1021 + 0.4008i w65  0.9006 − 0.4173i −0.8567 + 0.4063i w66  0.5578 − 0.4216i −0.5499 + 0.4932i w67  0.6797 − 0.3850i −0.6209 + 0.3666i w68  1.1837 − 0.8001i −0.1083 + 0.5383i w69  0.9008 − 0.6972i −0.1008 + 0.3768i w70  0.5354 − 0.6351i −0.3317 + 0.5081i w71  0.6549 − 0.7546i −0.3204 + 0.3654i w72  1.1598 − 0.1437i −1.0961 + 0.1322i w73  0.9102 − 0.1324i −0.8513 + 0.1298i w74  0.5755 − 0.1222i −0.5343 + 0.0862i w75  0.6787 − 0.1401i −0.6260 + 0.1694i w76  1.4855 − 0.1793i −0.1082 + 0.0775i w77  1.5681 − 0.5636i −0.1037 + 0.2076i w78  0.9727 − 1.2109i −0.3325 + 0.0910i w79  0.7102 − 1.0171i −0.3189 + 0.2137i w80  0.1045 − 0.4578i −1.1805 + 0.7084i w81  0.1067 − 0.3515i −0.8930 + 0.6876i w82  0.3411 − 0.4189i −0.5622 + 0.6728i w83  0.3088 − 0.3456i −0.6763 + 0.8371i w84  0.1121 − 0.6749i −0.1170 + 0.7305i w85  0.1179 − 0.8955i −0.1259 + 0.9453i w86  0.3342 − 0.6628i −0.3446 + 0.7129i w87  0.3585 − 0.8699i −0.3857 + 0.9233i w88  0.1075 − 0.0988i −1.4038 + 0.1588i w89  0.1136 − 0.1402i −1.5205 + 0.4861i w90  0.3700 − 0.1151i −1.0743 + 1.1040i w91  0.3294 − 0.1448i −0.7513 + 1.0912i w92  0.1933 − 1.4991i −0.1870 + 1.5043i w93  0.1440 − 1.1691i −0.1454 + 1.1973i w94  0.5835 − 1.4250i −0.5736 + 1.4545i w95  0.4344 − 1.1140i −0.4433 + 1.1606i w96 −1.1438 − 0.4323i −1.1021 − 0.4008i w97 −0.9006 − 0.4173i −0.8567 − 0.4063i w98 −0.5578 − 0.4216i −0.5499 − 0.4932i w99 −0.6797 − 0.3850i −0.6209 − 0.3666i w100 −1.1837 − 0.8001i −0.1083 − 0.5383i w101 −0.9008 − 0.6972i −0.1008 − 0.3768i w102 −0.5354 − 0.6351i −0.3317 − 0.5081i w103 −0.6549 − 0.7546i −0.3204 − 0.3654i w104 −1.1598 − 0.1437i −1.0961 − 0.1322i w105 −0.9102 − 0.1324i −0.8513 − 0.1298i w106 −0.5755 − 0.1222i −0.5343 − 0.0862i w107 −0.6787 − 0.1401i −0.6260 − 0.1694i w108 −1.4855 − 0.1793i −0.1082 − 0.0775i w109 −1.5681 − 0.5636i −0.1037 − 0.2076i w110 −0.9727 − 1.2109i −0.3325 − 0.0910i w111 −0.7102 − 1.0171i −0.3189 − 0.2137i w112 −0.1045 − 0.4578i −1.1805 − 0.7084i w113 −0.1067 − 0.3515i −0.8930 − 0.6876i w114 −0.3411 − 0.4189i −0.5622 − 0.6728i w115 −0.3088 − 0.3456i −0.6763 − 0.8371i w116 −0.1121 − 0.6749i −0.1170 − 0.7305i w117 −0.1179 − 0.8955i −0.1259 − 0.9453i w118 −0.3342 − 0.6628i −0.3446 − 0.7129i w119 −0.3585 − 0.8699i −0.3857 − 0.9233i w120 −0.1075 − 0.0988i −1.4038 − 0.1588i w121 −0.1136 − 0.1402i −1.5205 − 0.4861i w122 −0.3700 − 0.1151i −1.0743 − 1.1040i w123 −0.3294 − 0.1448i −0.7513 − 1.0912i w124 −0.1933 − 1.4991i −0.1870 − 1.5043i w125 −0.1440 − 1.1691i −0.1454 − 1.1973i w126 −0.5835 − 1.4250i −0.5736 − 1.4545i w127 −0.4344 − 1.1140i −0.4433 − 1.1606i E6) 256-QAM 2D NUC

code rate w 20/30 22/30 w0  1.6350 + 0.1593i  1.5977 + 0.1526i w1  1.5776 + 0.4735i  1.3187 + 0.1269i w2  0.9430 + 0.1100i −1.5977 + 0.1526i w3  0.9069 + 0.2829i −1.3187 + 0.1269i w4  0.3237 + 0.0849i  0.2574 + 0.0733i w5  0.3228 + 0.0867i  0.4496 + 0.0807i w6  0.7502 + 0.1138i −0.2574 + 0.0733i w7  0.7325 + 0.2088i −0.4496 + 0.0807i w8  0.1658 + 1.6747i  1.5977 − 0.1526i w9  0.4907 + 1.6084i  1.3187 − 0.1269i w10  0.1088 + 0.9530i −1.5977 − 0.1526i w11  0.2464 + 0.9270i −1.3187 − 0.1269i w12  0.0872 + 0.1390i  0.2574 − 0.0733i w13  0.0871 + 0.1392i  0.4496 − 0.0807i w14  0.1091 + 0.7656i −0.2574 − 0.0733i w15  0.1699 + 0.7537i −0.4496 − 0.0807i w16 −1.6350 + 0.1593i  0.9269 + 0.0943i w17 −1.5776 + 0.4735i  1.1024 + 0.1086i w18 −0.9430 + 0.1100i −0.9269 + 0.0943i w19 −0.9069 + 0.2829i −1.1024 + 0.1086i w20 −0.3237 + 0.0849i  0.7663 + 0.0867i w21 −0.3228 + 0.0867i  0.6115 + 0.0871i w22 −0.7502 + 0.1138i −0.7663 + 0.0867i w23 −0.7325 + 0.2088i −0.6115 + 0.0871i w24 −0.1658 + 1.6747i  0.9269 − 0.0943i w25 −0.4907 + 1.6084i  1.1024 − 0.1086i w26 −0.1088 + 0.9530i −0.9269 − 0.0943i w27 −0.2464 + 0.9270i −1.1024 − 0.1086i w28 −0.0872 + 0.1390i  0.7663 − 0.0867i w29 −0.0871 + 0.1392i  0.6115 − 0.0871i w30 −0.1091 + 0.7656i −0.7663 − 0.0867i w31 −0.1699 + 0.7537i −0.6115 − 0.0871i w32  1.3225 + 0.1320i  1.2701 + 1.0139i w33  1.2742 + 0.3922i  1.0525 + 0.8406i w34  1.0854 + 0.1139i −1.2701 + 1.0139i w35  1.0441 + 0.3296i −1.0525 + 0.8406i w36  0.4582 + 0.1123i  0.2487 + 0.1978i w37  0.4545 + 0.1251i  0.3523 + 0.2915i w38  0.6473 + 0.1138i −0.2487 + 0.1978i w39  0.6339 + 0.1702i −0.3523 + 0.2915i w40  0.1322 + 1.3631i  1.2701 − 1.0139i w41  0.3929 + 1.3102i  1.0525 − 0.8406i w42  0.1124 + 1.1327i −1.2701 − 1.0139i w43  0.3160 + 1.0913i −1.0525 − 0.8406i w44  0.0928 + 0.3970i  0.2487 − 0.1978i w45  0.0937 + 0.3973i  0.3523 − 0.2915i w46  0.1054 + 0.5979i −0.2487 − 0.1978i w47  0.1230 + 0.5949i −0.3523 − 0.2915i w48 −1.3225 + 0.1320i  0.7359 + 0.6043i w49 −1.2742 + 0.3922i  0.8807 + 0.7105i w50 −1.0854 + 0.1139i −0.7359 + 0.6043i w51 −1.0441 + 0.3296i −0.8807 + 0.7105i w52 −0.4582 + 0.1123i  0.6017 + 0.5019i w53 −0.4545 + 0.1251i  0.4747 + 0.3996i w54 −0.6473 + 0.1138i −0.6017 + 0.5019i w55 −0.6339 + 0.1702i −0.4747 + 0.3996i w56 −0.1322 + 1.3631i  0.7359 − 0.6043i w57 −0.3929 + 1.3102i  0.8807 − 0.7105i w58 −0.1124 + 1.1327i −0.7359 − 0.6043i w59 −0.3160 + 1.0913i −0.8807 − 0.7105i w60 −0.0928 + 0.3970i  0.6017 − 0.5019i w61 −0.0937 + 0.3973i  0.4747 − 0.3996i w62 −0.1054 + 0.5979i −0.6017 − 0.5019i w63 −0.1230 + 0.5949i −0.4747 − 0.3996i w64  1.6350 − 0.1593i  1.5441 + 0.4545i w65  1.5776 − 0.4735i  1.2750 + 0.3775i w66  0.9430 − 0.1100i −1.5441 + 0.4545i w67  0.9069 − 0.2829i −1.2750 + 0.3775i w68  0.3237 − 0.0849i  0.2586 + 0.0752i w69  0.3228 − 0.0867i  0.4435 + 0.1065i w70  0.7502 − 0.1138i −0.2586 + 0.0752i w71  0.7325 − 0.2088i −0.4435 + 0.1065i w72  0.1658 − 1.6747i  1.5441 − 0.4545i w73  0.4907 − 1.6084i  1.2750 − 0.3775i w74  0.1088 − 0.9530i −1.5441 − 0.4545i w75  0.2464 − 0.9270i −1.2750 − 0.3775i w76  0.0872 − 0.1390i  0.2586 − 0.0752i w77  0.0871 − 0.1392i  0.4435 − 0.1065i w78  0.1091 − 0.7656i −0.2586 − 0.0752i w79  0.1699 − 0.7537i −0.4435 − 0.1065i w80 −1.6350 − 0.1593i  0.8925 + 0.2771i w81 −1.5776 − 0.4735i  1.0649 + 0.3219i w82 −0.9430 − 0.1100i −0.8925 + 0.2771i w83 −0.9069 − 0.2829i −1.0649 + 0.3219i w84 −0.3237 − 0.0849i  0.7362 + 0.2279i w85 −0.3228 − 0.0867i  0.5936 + 0.1699i w86 −0.7502 − 0.1138i −0.7362 + 0.2279i w87 −0.7325 − 0.2088i −0.5936 + 0.1699i w88 −0.1658 − 1.6747i  0.8925 − 0.2771i w89 −0.4907 − 1.6084i  1.0649 − 0.3219i w90 −0.1088 − 0.9530i −0.8925 − 0.2771i w91 −0.2464 − 0.9270i −1.0649 − 0.3219i w92 −0.0872 − 0.1390i  0.7362 − 0.2279i w93 −0.0871 − 0.1392i  0.5936 − 0.1699i w94 −0.1091 − 0.7656i −0.7362 − 0.2279i w95 −0.1699 − 0.7537i −0.5936 − 0.1699i w96  1.3225 − 0.1320i  1.4352 + 0.7452i w97  1.2742 − 0.3922i  1.1866 + 0.6182i w98  1.0854 − 0.1139i −1.4352 + 0.7452i w99  1.0441 − 0.3296i −1.1866 + 0.6182i w100  0.4582 − 0.1123i  0.2523 + 0.1944i w101  0.4545 − 0.1251i  0.3695 + 0.2695i w102  0.6473 − 0.1138i −0.2523 + 0.1944i w103  0.6339 − 0.1702i −0.3695 + 0.2695i w104  0.1322 − 1.3631i  1.4352 − 0.7452i w105  0.3929 − 1.3102i  1.1866 − 0.6182i w106  0.1124 − 1.1327i −1.4352 − 0.7452i w107  0.3160 − 1.0913i −1.1866 − 0.6182i w108  0.0928 − 0.3970i  0.2523 − 0.1944i w109  0.0937 − 0.3973i  0.3695 − 0.2695i w110  0.1054 − 0.5979i −0.2523 − 0.1944i w111  0.1230 − 0.5949i −0.3695 − 0.2695i w112 −1.3225 − 0.1320i  0.8273 + 0.4493i w113 −1.2742 − 0.3922i  0.9911 + 0.5243i w114 −1.0854 − 0.1139i −0.8273 + 0.4493i w115 −1.0441 − 0.3296i −0.9911 + 0.5243i w116 −0.4582 − 0.1123i  0.6708 + 0.3859i w117 −0.4545 − 0.1251i  0.5197 + 0.3331i w118 −0.6473 − 0.1138i −0.6708 + 0.3859i w119 −0.6339 − 0.1702i −0.5197 + 0.3331i w120 −0.1322 − 1.3631i  0.8273 − 0.4493i w121 −0.3929 − 1.3102i  0.9911 − 0.5243i w122 −0.1124 − 1.1327i −0.8273 − 0.4493i w123 −0.3160 − 1.0913i −0.9911 − 0.5243i w124 −0.0928 − 0.3970i  0.6708 − 0.3859i w125 −0.0937 − 0.3973i  0.5197 − 0.3331i w126 −0.1054 − 0.5979i −0.6708 − 0.3859i w127 −0.1230 − 0.5949i −0.5197 − 0.3331i w128  1.2901 + 1.0495i  0.1646 + 1.6329i w129  1.4625 + 0.7740i  0.1379 + 1.3595i w130  0.7273 + 0.6160i −0.1646 + 1.6329i w131  0.8177 + 0.4841i −0.1379 + 1.3595i w132  0.2844 + 0.1296i  0.0736 + 0.0898i w133  0.2853 + 0.1309i  0.0742 + 0.5054i w134  0.5902 + 0.4857i −0.0736 + 0.0898i w135  0.6355 + 0.4185i −0.0742 + 0.5054i w136  1.0646 + 1.2876i  0.1646 − 1.6329i w137  0.7949 + 1.4772i  0.1379 − 1.3595i w138  0.5707 + 0.7662i −0.1646 − 1.6329i w139  0.4490 + 0.8461i −0.1379 − 1.3595i w140  0.1053 + 0.1494i  0.0736 − 0.0898i w141  0.1052 + 0.1495i  0.0742 − 0.5054i w142  0.4294 + 0.6363i −0.0736 − 0.0898i w143  0.3744 + 0.6744i −0.0742 − 0.5054i w144 −1.2901 + 1.0495i  0.0992 + 0.9847i w145 −1.4625 + 0.7740i  0.1170 + 1.1517i w146 −0.7273 + 0.6160i −0.0992 + 0.9847i w147 −0.8177 + 0.4841i −0.1170 + 1.1517i w148 −0.2844 + 0.1296i  0.0894 + 0.8287i w149 −0.2853 + 0.1309i  0.0889 + 0.6739i w150 −0.5902 + 0.4857i −0.0894 + 0.8287i w151 −0.6355 + 0.4185i −0.0889 + 0.6739i w152 −1.0646 + 1.2876i  0.0992 − 0.9847i w153 −0.7949 + 1.4772i  0.1170 − 1.1517i w154 −0.5707 + 0.7662i −0.0992 − 0.9847i w155 −0.4490 + 0.8461i −0.1170 − 1.1517i w156 −0.1053 + 0.1494i  0.0894 − 0.8287i w157 −0.1052 + 0.1495i  0.0889 − 0.6739i w158 −0.4294 + 0.6363i −0.0894 − 0.8287i w159 −0.3744 + 0.6744i −0.0889 − 0.6739i w160  1.0382 + 0.8623i  1.0516 + 1.2481i w161  1.1794 + 0.6376i  0.8742 + 1.0355i w162  0.8504 + 0.7217i −1.0516 + 1.2481i w163  0.9638 + 0.5407i −0.8742 + 1.0355i w164  0.3734 + 0.2560i  0.0970 + 0.2450i w165  0.3799 + 0.2517i  0.1959 + 0.4045i w166  0.4968 + 0.3947i −0.0970 + 0.2450i w167  0.5231 + 0.3644i −0.1959 + 0.4045i w168  0.8555 + 1.0542i  1.0516 − 1.2481i w169  0.6363 + 1.2064i  0.8742 − 1.0355i w170  0.6961 + 0.8850i −1.0516 − 1.2481i w171  0.5229 + 1.0037i −0.8742 − 1.0355i w172  0.1938 + 0.3621i  0.0970 − 0.2450i w173  0.1909 + 0.3627i  0.1959 − 0.4045i w174  0.3224 + 0.5236i −0.0970 − 0.2450i w175  0.3016 + 0.5347i −0.1959 − 0.4045i w176 −1.0382 + 0.8623i  0.6150 + 0.7441i w177 −1.1794 + 0.6376i  0.7345 + 0.8743i w178 −0.8504 + 0.7217i −0.6150 + 0.7441i w179 −0.9638 + 0.5407i −0.7345 + 0.8743i w180 −0.3734 + 0.2560i  0.4932 + 0.6301i w181 −0.3799 + 0.2517i  0.3620 + 0.5258i w182 −0.4968 + 0.3947i −0.4932 + 0.6301i w183 −0.5231 + 0.3644i −0.3620 + 0.5258i w184 −0.8555 + 1.0542i  0.6150 − 0.7441i w185 −0.6363 + 1.2064i  0.7345 − 0.8743i w186 −0.6961 + 0.8850i −0.6150 − 0.7441i w187 −0.5229 + 1.0037i −0.7345 − 0.8743i w188 −0.1938 + 0.3621i  0.4932 − 0.6301i w189 −0.1909 + 0.3627i  0.3620 − 0.5258i w190 −0.3224 + 0.5236i −0.4932 − 0.6301i w191 −0.3016 + 0.5347i −0.3620 − 0.5258i w192  1.2901 − 1.0495i  0.4866 + 1.5660i w193  1.4625 − 0.7740i  0.4068 + 1.3027i w194  0.7273 − 0.6160i −0.4866 + 1.5660i w195  0.8177 − 0.4841i −0.4068 + 1.3027i w196  0.2844 − 0.1296i  0.0732 + 0.0899i w197  0.2853 − 0.1309i  0.0877 + 0.4997i w198  0.5902 − 0.4857i −0.0732 + 0.0899i w199  0.6355 − 0.4185i −0.0877 + 0.4997i w200  1.0646 − 1.2876i  0.4866 − 1.5660i w201  0.7949 − 1.4772i  0.4068 − 1.3027i w202  0.5707 − 0.7662i −0.4866 − 1.5660i w203  0.4490 − 0.8461i −0.4068 − 1.3027i w204  0.1053 − 0.1494i  0.0732 − 0.0899i w205  0.1052 − 0.1495i  0.0877 − 0.4997i w206  0.4294 − 0.6363i −0.0732 − 0.0899i w207  0.3744 − 0.6744i −0.0877 − 0.4997i w208 −1.2901 − 1.0495i  0.2927 + 0.9409i w209 −1.4625 − 0.7740i  0.3446 + 1.1023i w210 −0.7273 − 0.6160i −0.2927 + 0.9409i w211 −0.8177 − 0.4841i −0.3446 + 1.1023i w212 −0.2844 − 0.1296i  0.2350 + 0.7945i w213 −0.2853 − 0.1309i  0.1670 + 0.6529i w214 −0.5902 − 0.4857i −0.2350 + 0.7945i w215 −0.6355 − 0.4185i −0.1670 + 0.6529i w216 −1.0646 − 1.2876i  0.2927 − 0.9409i w217 −0.7949 − 1.4772i  0.3446 − 1.1023i w218 −0.5707 − 0.7662i −0.2927 − 0.9409i w219 −0.4490 − 0.8461i −0.3446 − 1.1023i w220 −0.1053 − 0.1494i  0.2350 − 0.7945i w221 −0.1052 − 0.1495i  0.1670 − 0.6529i w222 −0.4294 − 0.6363i −0.2350 − 0.7345i w223 −0.3744 − 0.6744i −0.1670 − 0.6529i w224  1.0382 − 0.8623i  0.7867 + 1.4356i w225  1.1794 − 0.6376i  0.6561 + 1.1927i w226  0.8504 − 0.7217i −0.7867 + 1.4356i w227  0.9638 − 0.5407i −0.6561 + 1.1927i w228  0.3734 − 0.2560i  0.0947 + 0.2451i w229  0.3799 − 0.2517i  0.1865 + 0.4121i w230  0.4968 − 0.3947i −0.0947 + 0.2451i w231  0.5231 − 0.3644i −0.1865 + 0.4121i w232  0.8555 − 1.0542i  0.7867 − 1.4356i w233  0.6363 − 1.2064i  0.6561 − 1.1927i w234  0.6961 − 0.8850i −0.7867 − 1.4356i w235  0.5229 − 1.0037i −0.6561 − 1.1927i w236  0.1938 − 0.3621i  0.0947 − 0.2451i w237  0.1909 − 0.3627i  0.1865 − 0.4121i w238  0.3224 − 0.5235i −0.0947 − 0.2451i w239  0.3016 − 0.5347i −0.1865 − 0.4121i w240 −1.0382 − 0.8623i  0.4677 + 0.8579i w241 −1.1794 − 0.6376i  0.5537 + 1.0081i w242 −0.8504 − 0.7217i −0.4677 + 0.8579i w243 −0.9638 − 0.5407i −0.5537 + 1.0081i w244 −0.3734 − 0.2560i  0.3893 + 0.7143i w245 −0.3799 − 0.2517i  0.3110 + 0.5686i w246 −0.4968 − 0.3947i −0.3893 + 0.7143i w247 −0.5231 − 0.3644i −0.3110 + 0.5686i w248 −0.8555 − 1.0542i  0.4677 − 0.8579i w249 −0.6363 − 1.2064i  0.5537 − 1.0081i w250 −0.6961 − 0.8850i −0.4677 − 0.8579i w251 −0.5229 − 1.0037i −0.5537 − 1.0081i w252 −0.1938 − 0.3621i  0.3893 − 0.7143i w253 −0.1909 − 0.3627i  0.3110 − 0.5686i w254 −0.3224 − 0.5236i −0.3893 − 0.7143i w255 −0.3016 − 0.5347i −0.3110 − 0.5686i

Constellations from group E can preferably be used for coding and modulation in accordance with the DVB-S2 standard or its extension DVB-Sx. Special care has been taken such that the constellations can be used together with LDPC codes and bit interleaver settings from the DVB-Sx baseline system.

Whenever constellations are proposed for LDPC codes not yet part of the DVB-Sx baseline, like e.g. codes of code rate “x/30”, the constellations are optimized such that they allow for a DVB-S2-like bit interleaver. This means, the same interleaving rule can be applied as used in DVB-S2 for 64 k LDPC codes (except for code rate 3/5 from S2): the block interleaver is filled column-wise, and read out row-wise, each row read from left to right. In terms of bit interleaver patterns, as discussed in the DVB-Sx baseline, this would correspond to bit interleaver patterns [0,1,2, . . . M−1], in which M is the number of bits/QAM symbol, e.g., M=3 for a constellation with 8 points, M=4, for 16 points, and so on.

It should be noted that the code rates for which the constellations of groups A, B, C, D and E have been optimized are the code rates of the LDPC encoder. However, the total code rates may actually be smaller due to the use of an additional BCH encoder.

FIG. 13 shows a diagram for a constellation from group E for 8-NUC for code rate R=100/180 or 104/180. The corresponding constellation point vector is w=(−0.2330, −0.5414−0.8712i, −0.5414+0.8712i, −1.3570, 0.2330, 0.5414−0.8712i, 0.5414+0.8712i, 1.3570).

FIG. 14 shows a diagram illustrating the shaping gain for the above explained non-uniform 8-ary QAM constellations which may be used for a standard relating to digital video broadcasting via satellite. The curves were obtained applying the following conditions: AWGN channel, bit error rate and frame error rates (BER. FER) after 50 LDPC iterations, and outer BCH decoding; 64 k LDPC codes were used (code length=64800). The diagram includes curves showing the BER (solid lines) and the FER (dashed lines). Further, there are curves for conventional constellations as used in known standards using baseline constellations and curves for the above explained NUCs for 8-ary QAMs. It can be seen that the same BER or FER can be achieved at lower SNR. The best result is obtained for code rate 13/18: the proposed constellation works at 0.55 dB lower SNR than the conventional constellation. The curves are almost shifted in parallel to the left with the optimized constellation by this offset. For code rate 25/36 a 0.5 dB gain is achieved.

FIGS. 15AA-15CB show examples of LDPC codes for which the disclosed constellations have been optimized. The code length and the code rate are given for the respective codes.

It should be noted that the present disclosure is to be understood such that the disclosure includes embodiments of coding and modulation apparatus for which less groups of tables of constellations are available for selection and/or use of a constellation, for which smaller tables of constellations are available for selection and/or use of a constellation, for which tables including constellations for less code rates and/or less values of M are available for selection and/or use of a constellation, and/or for which only selected (single) constellations from among all the disclosed constellations are available for selection and/or use of a constellation.

Obviously, numerous modifications and variations of the present disclosure are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the disclosure may be practiced otherwise than as specifically described herein (e.g., if the NUC position vectors are rounded to a smaller number of digits).

In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

In so far as embodiments of the disclosure have been described as being implemented, at least in part, by software-controlled data processing apparatus, it will be appreciated that a non-transitory machine-readable medium carrying such software, such as an optical disk, a magnetic disk, semiconductor memory or the like, is also considered to represent an embodiment of the present disclosure. Further, such a software may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems.

A circuit that may be used for implementing one or more of the elements of the claimed apparatus is a structural assemblage of electronic components including conventional circuit elements, integrated circuits including application specific integrated circuits, standard integrated circuits, application specific standard products, and field programmable gate arrays. Further a circuit includes central processing units, graphics processing units, and microprocessors which are programmed or configured according to software code. A circuit does not include pure software, although a circuit includes the above-described hardware executing software.

Any reference signs in the claims should not be construed as limiting the scope. 

The invention claimed is:
 1. A receiving apparatus comprising: a receiver configured to receive at least one transmission stream including symbols representative of constellation values; and at least one processor configured to demodulate the symbols into 10-bit cell words and to decode the 10-bit cell words into output data words representing Forward Error Correction (FEC) data words, wherein each 10-bit cell word is associated with a point of a Non-Uniform Constellation (NUC), wherein each point (Zq) of the NUC has an in-phase component Re(Zq) and a quadrature component Im(Zq), wherein the in-phase component Re(Zq) corresponds to a constellation position of a first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), wherein the quadrature component Im(Zq) corresponds to a constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), and wherein the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15) corresponds to a vector prior to power normalization defined as (1, 1.0003, 1.0149, 1.0158, 2.6848, 2.6903, 2.882, 2.8747, 4.7815, 4.7619, 5.5779, 5.6434, 7.3854, 7.8797, 9.635, 11.7874).
 2. The receiving apparatus according to claim 1, wherein a code rate of the FEC is 6/15.
 3. The receiving apparatus according to claim 1, wherein the at least one transmission stream is a digital video broadcasting stream.
 4. The receiving apparatus according to claim 3, wherein the at least one transmission stream is a terrestrial digital video broadcasting stream.
 5. The receiving apparatus according to claim 1, wherein the NUC is a Quadrature Amplitude Modulation (QAM) constellation.
 6. The receiving apparatus according to claim 1, wherein bits of the 10-bit cell words are labeled on the NUC in accordance with a binary reflected Gray mapping.
 7. The receiving apparatus according to claim 1, wherein respective in-phase and quadrature positions each provide 5 bits of the 10-bit cell words, and wherein the in-phase and quadrature positions together provide ten bits (y_(0,q), y_(1,q); . . . y_(9,q)) of the 10-bit cell words, a first set of five bits being provided by the in phase position and a second set of five bits being provided by the quadrature position.
 8. The receiving apparatus according to claim 7, wherein the first set of five bits includes bits (y_(0q), y_(2q), y_(4q), y_(6q), y_(8q)) and the second set of five bits includes (y_(1q), y_(3q), y_(5q), y_(7q), y_(9q)).
 9. The receiving apparatus according to claim 7, wherein at least one of the first set of five bits and the second set of five bits is provided in accordance with bit labels of the following table: position −u15 providing bit label 1 0 0 0 0 position −u14 providing bit label 1 0 0 0 1 position −u13 providing bit label 1 0 0 1 1 position −u12 providing bit label 1 0 0 1 0 position −u11 providing bit label 1 0 1 1 0 position −u10 providing bit label 1 0 1 1 1 position −u9 providing bit label 1 0 1 0 1 position −u8 providing bit label 1 0 1 0 0 position −u7 providing bit label 1 1 1 0 0 position −u6 providing bit label 1 1 1 0 1 position −u5 providing bit label 1 1 1 1 1 position −u4 providing bit label 1 1 1 1 0 position −u3 providing bit label 1 1 0 1 0 position −u2 providing bit label 1 1 0 1 1 position −u1 providing bit label 1 1 0 0 1 position −u0 providing bit label 1 1 0 0 0 position u0 providing bit label 0 1 0 0 0 position u1 providing bit label 0 1 0 0 1 position u2 providing bit label 0 1 0 1 1 position u3 providing bit label 0 1 0 1 0 position u4 providing bit label 0 1 1 1 0 position u5 providing bit label 0 1 1 1 1 position u6 providing bit label 0 1 1 0 1 position u7 providing bit label 0 1 1 0 0 position u8 providing bit label 0 0 1 0 0 position u9 providing bit label 0 0 1 0 1 position u10 providing bit label 0 0 1 1 1 position u11 providing bit label 0 0 1 1 0 position u12 providing bit label 0 0 0 1 0 position u13 providing bit label 0 0 0 1 1 position u14 providing bit label 0 0 0 0 1 position u15 providing bit label 0 0 0 0
 0. 10. The receiving apparatus according to claim 1, wherein the at least one processor is configured to select the NUC in accordance with a code rate of the FEC.
 11. The receiving apparatus according to claim 1, wherein the in-phase component Re(Zq) corresponds to a rounded normalized constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), and wherein the quadrature component Im(Zq) corresponds to a rounded normalized constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15).
 12. A receiving method comprising: receiving at least one transmission stream including symbols representative of constellation values; demodulating the symbols into 10-bit cell words and to decode the 10-bit cell words into output data words representing Forward Error Correction (FEC) data words, wherein each 10-bit cell word is associated with a point of a Non-Uniform Constellation (NUC), wherein each point (Zq) of the NUC has an in-phase component Re(Zq) and a quadrature component Im(Zq), wherein the in-phase component Re(Zq) corresponds to a constellation position of a first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), wherein the quadrature component Im(Zq) corresponds to a constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), and wherein the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10 u11, u12, u13, u14, u15) corresponds to a vector prior to power normalization defined as (1, 1.0003, 1.0149, 1.0158, 2.6848, 2.6903, 2.882, 2.8747, 4.7815, 4.7619, 5.5779, 5.6434, 7.3854, 7.8797, 9.635, 11.7874).
 13. The receiving method according to claim 12, wherein a code rate of the forward error correction is 6/15.
 14. The receiving method according to claim 12, wherein the at least one transmission stream is a digital video broadcasting stream.
 15. The receiving method according to claim 12, wherein bits of the 10-bit cell words are labeled on the NUC in accordance with a binary reflected Gray mapping.
 16. The receiving method according to claim 12, wherein respective in-phase and quadrature positions each provide 5 bits of the 10-bit cell words, and wherein the in-phase and quadrature positions together provide ten bits (y_(0,q), y_(1,q); . . . y_(9,q)) of the 10-bit cell words, a first set of five bits being provided by the in phase position and a second set of five bits being provided by the quadrature position.
 17. The receiving method according to claim 16, wherein at least one of the first set of five bits and the second set of five bits is provided in accordance with bit labels of the following table: position −u15 providing bit label 1 0 0 0 0 position −u14 providing bit label 1 0 0 0 1 position −u13 providing bit label 1 0 0 1 1 position −u12 providing bit label 1 0 0 1 0 position −u11 providing bit label 1 0 1 1 0 position −u10 providing bit label 1 0 1 1 1 position −u9 providing bit label 1 0 1 0 1 position −u8 providing bit label 1 0 1 0 0 position −u7 providing bit label 1 1 1 0 0 position −u6 providing bit label 1 1 1 0 1 position −u5 providing bit label 1 1 1 1 1 position −u4 providing bit label 1 1 1 1 0 position −u3 providing bit label 1 1 0 1 0 position −u2 providing bit label 1 1 0 1 1 position −u1 providing bit label 1 1 0 0 1 position −u0 providing bit label 1 1 0 0 0 position u0 providing bit label 0 1 0 0 0 position u1 providing bit label 0 1 0 0 1 position u2 providing bit label 0 1 0 1 1 position u3 providing bit label 0 1 0 1 0 position u4 providing bit label 0 1 1 1 0 position u5 providing bit label 0 1 1 1 1 position u6 providing bit label 0 1 1 0 1 position u7 providing bit label 0 1 1 0 0 position u8 providing bit label 0 0 1 0 0 position u9 providing bit label 0 0 1 0 1 position u10 providing bit label 0 0 1 1 1 position u11 providing bit label 0 0 1 1 0 position u12 providing bit label 0 0 0 1 0 position u13 providing bit label 0 0 0 1 1 position u14 providing bit label 0 0 0 0 1 position u15 providing bit label 0 0 0 0
 0. 18. The receiving method according to claim 12, wherein the in-phase component Re(Zq) corresponds to a rounded normalized constellation position of a first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), and wherein the quadrature component Im(Zq) corresponds to a rounded normalized constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15).
 19. A non-transitory computer-readable storage medium storing computer-readable instructions, which, when executed by a receiving apparatus, cause the receiving apparatus to perform a receiving method comprising: receiving at least one transmission stream including symbols representative of constellation values; demodulating the symbols into 10-bit cell words and to decode the 10-bit cell words into output data words representing Forward Error Correction (FEC) data words, wherein each 10-bit cell word is associated with a point of a Non-Uniform Constellation (NUC), wherein each point (Zq) of the NUC has an in-phase component Re(Zq) and a quadrature component Im(Zq), wherein the in-phase component Re(Zq) corresponds to a constellation position of a first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), wherein the quadrature component Im(Zq) corresponds to a constellation position of the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15), and wherein the first vector (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10, u11, u12, u13, u14, u15) corresponds to a vector prior to power normalization defined as (1, 1.0003, 1.0149, 1.0158, 2.6848, 2.6903, 2.882, 2.8747, 4.7815, 4.7619, 5.5779, 5.6434, 7.3854, 7.8797, 9.635, 11.7874). 